Ring homomorphisms
142-464: Amalie Emmy Noether ( US : / ˈ n ʌ t ər / , UK : / ˈ n ɜː t ə / ; German: [ˈnøːtɐ] ; 23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra . She proved Noether's first and second theorems , which are fundamental in mathematical physics . She was described by Pavel Alexandrov , Albert Einstein , Jean Dieudonné , Hermann Weyl and Norbert Wiener as
284-420: A {\displaystyle {\mathfrak {a}}} and b {\displaystyle {\mathfrak {b}}} , left (resp. right) ideals of a ring R , their sum is which is a left (resp. right) ideal, and, if a , b {\displaystyle {\mathfrak {a}},{\mathfrak {b}}} are two-sided, i.e. the product is the ideal generated by all products of
426-411: A ⊂ b {\displaystyle {\mathfrak {a}}\subset {\mathfrak {b}}} , there is an ideal c {\displaystyle {\mathfrak {c}}} such that a = b c {\displaystyle {\mathfrak {\mathfrak {a}}}={\mathfrak {b}}{\mathfrak {c}}} . It can then be shown that every nonzero ideal of
568-444: A ⊋ Ann ( J n ) {\displaystyle {\mathfrak {a}}\supsetneq \operatorname {Ann} (J^{n})} is an ideal properly minimal over the latter, then J ⋅ ( a / Ann ( J n ) ) = 0 {\displaystyle J\cdot ({\mathfrak {a}}/\operatorname {Ann} (J^{n}))=0} . That is, J n
710-481: A , b , c {\displaystyle {\mathfrak {a}},{\mathfrak {b}},{\mathfrak {c}}} , If a product is replaced by an intersection, a partial distributive law holds: where the equality holds if a {\displaystyle {\mathfrak {a}}} contains b {\displaystyle {\mathfrak {b}}} or c {\displaystyle {\mathfrak {c}}} . Remark : The sum and
852-386: A , R / b ) = ( a ∩ b ) / a b {\displaystyle \operatorname {Tor} _{1}^{R}(R/{\mathfrak {a}},R/{\mathfrak {b}})=({\mathfrak {a}}\cap {\mathfrak {b}})/{\mathfrak {a}}{\mathfrak {b}}} . ) An integral domain is called a Dedekind domain if for each pair of ideals
994-497: A = J n + 1 a = 0 {\displaystyle J^{n}{\mathfrak {a}}=J^{n+1}{\mathfrak {a}}=0} , a contradiction.) Let A and B be two commutative rings , and let f : A → B be a ring homomorphism . If a {\displaystyle {\mathfrak {a}}} is an ideal in A , then f ( a ) {\displaystyle f({\mathfrak {a}})} need not be an ideal in B (e.g. take f to be
1136-416: A = ( z , w ) , b = ( x + z , y + w ) , c = ( x + z , w ) {\displaystyle {\mathfrak {a}}=(z,w),{\mathfrak {b}}=(x+z,y+w),{\mathfrak {c}}=(x+z,w)} . Then, In the first computation, we see the general pattern for taking the sum of two finitely generated ideals, it is the ideal generated by
1278-592: A Realgymnasium in Nuremberg . During the 1903–1904 winter semester, she studied at the University of Göttingen , attending lectures given by astronomer Karl Schwarzschild and mathematicians Hermann Minkowski , Otto Blumenthal , Felix Klein , and David Hilbert . In 1903, restrictions on women's full enrollment in Bavarian universities were rescinded. Noether returned to Erlangen and officially reentered
1420-707: A cot–caught merger , which is rapidly spreading throughout the whole country. However, the South, Inland North, and a Northeastern coastal corridor passing through Rhode Island, New York City, Philadelphia, and Baltimore typically preserve an older cot–caught distinction. For that Northeastern corridor, the realization of the THOUGHT vowel is particularly marked , as depicted in humorous spellings, like in tawk and cawfee ( talk and coffee ), which intend to represent it being tense and diphthongal : [oə] . A split of TRAP into two separate phonemes , using different
1562-502: A left ideal I is a subrng with the additional property that r x {\displaystyle rx} is in I for every r ∈ R {\displaystyle r\in R} and every x ∈ I {\displaystyle x\in I} . (Right and two-sided ideals are defined similarly.) For a ring, an ideal I (say a left ideal) is rarely a subring; since
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#17327730701001704-679: A pension lodging building, after student leaders complained of living with "a Marxist-leaning Jewess". Hermann Weyl recalled that "During the wild times after the Revolution of 1918 ," Noether "sided more or less with the Social Democrats ". She was from 1919 through 1922 a member of the Independent Social Democrats , a short-lived splinter party. In the words of logician and historian Colin McLarty , "she
1846-520: A pronunciations for example in gap [æ] versus gas [eə] , further defines New York City as well as Philadelphia–Baltimore accents. Most Americans preserve all historical /r/ sounds, using what is known as a rhotic accent . The only traditional r -dropping (or non-rhoticity) in regional U.S. accents variably appears today in eastern New England , New York City , and some of the former plantation South primarily among older speakers (and, relatedly, some African-American Vernacular English across
1988-412: A quotient group . Among the integers, the ideals correspond one-for-one with the non-negative integers : in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to
2130-623: A Dedekind domain can be uniquely written as a product of maximal ideals, a generalization of the fundamental theorem of arithmetic . In Z {\displaystyle \mathbb {Z} } we have since ( n ) ∩ ( m ) {\displaystyle (n)\cap (m)} is the set of integers that are divisible by both n {\displaystyle n} and m {\displaystyle m} . Let R = C [ x , y , z , w ] {\displaystyle R=\mathbb {C} [x,y,z,w]} and let
2272-482: A commutative ring R , then a ∩ b = a b {\displaystyle {\mathfrak {a}}\cap {\mathfrak {b}}={\mathfrak {a}}{\mathfrak {b}}} in the following two cases (at least) (More generally, the difference between a product and an intersection of ideals is measured by the Tor functor : Tor 1 R ( R /
2414-631: A complex phenomenon of "both convergence and divergence": some accents are homogenizing and leveling , while others are diversifying and deviating further away from one another. Having been settled longer than the American West Coast, the East Coast has had more time to develop unique accents, and it currently comprises three or four linguistically significant regions, each of which possesses English varieties both different from each other as well as quite internally diverse: New England ,
2556-447: A consonant, such as in pearl , car and fort . Non-rhotic American accents, those that do not pronounce ⟨r⟩ except before a vowel, such as some accents of Eastern New England , New York City , and African-Americans , and a specific few (often older ones) spoken by Southerners , are often quickly noticed by General American listeners and perceived as sounding especially ethnic, regional, or antiquated. Rhoticity
2698-405: A contradiction. Since a nonzero finitely generated module admits a maximal submodule, in particular, one has: A maximal ideal is a prime ideal and so one has where the intersection on the left is called the nilradical of R . As it turns out, nil ( R ) {\displaystyle \operatorname {nil} (R)} is also the set of nilpotent elements of R . If R
2840-591: A merger with the THOUGHT ( caught ) set. Having taken place prior to the unrounding of the cot vowel, it results in lengthening and perhaps raising, merging the more recently separated vowel into the THOUGHT vowel in the following environments: before many instances of /f/ , /θ/ , and particularly /s/ (as in Austria, cloth, cost, loss, off, often, etc.), a few instances before /ŋ/ (as in strong, long, wrong ), and variably by region or speaker in gone , on , and certain other words. Unlike American accents,
2982-636: A nice day , for sure); many are now distinctly old-fashioned (swell, groovy). Some English words now in general use, such as hijacking, disc jockey , boost, bulldoze and jazz , originated as American slang. American English has always shown a marked tendency to use words in different parts of speech and nouns are often used as verbs . Examples of nouns that are now also verbs are interview, advocate, vacuum, lobby, pressure, rear-end, transition, feature, profile, hashtag, head, divorce, loan, estimate, X-ray, spearhead, skyrocket, showcase, bad-mouth, vacation , major, and many others. Compounds coined in
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#17327730701003124-591: A noncommutative reciprocity law . This pleased her immensely. He also sent her a mathematical riddle, which he called the "m μν -riddle of syllables". She solved it immediately, but the riddle has been lost. In September of the same year, Noether delivered a plenary address ( großer Vortrag ) on "Hyper-complex systems in their relations to commutative algebra and to number theory" at the International Congress of Mathematicians in Zürich . The congress
3266-443: A not-necessarily-commutative ring, it is a general fact that 1 − y x {\displaystyle 1-yx} is a unit element if and only if 1 − x y {\displaystyle 1-xy} is (see the link) and so this last characterization shows that the radical can be defined both in terms of left and right primitive ideals. The following simple but important fact ( Nakayama's lemma )
3408-457: A note in the seventh edition "based in part on lectures by E. Artin and E. Noether". Beginning in 1927, Noether worked closely with Emil Artin , Richard Brauer and Helmut Hasse on noncommutative algebras . Van der Waerden's visit was part of a convergence of mathematicians from all over the world to Göttingen, which had become a major hub of mathematical and physical research. Russian mathematicians Pavel Alexandrov and Pavel Urysohn were
3550-429: A nurturing attitude. In Noether's obituary, Van der Waerden described her as Completely unegotistical and free of vanity, she never claimed anything for herself, but promoted the works of her students above all. Noether showed a devotion to her subject and her students that extended beyond the academic day. Once, when the building was closed for a state holiday, she gathered the class on the steps outside, led them through
3692-480: A process of extensive dialect mixture and leveling in which English varieties across the colonies became more homogeneous compared with the varieties in Britain. English thus predominated in the colonies even by the end of the 17th century's first immigration of non-English speakers from Western Europe and Africa. Additionally, firsthand descriptions of a fairly uniform American English (particularly in contrast to
3834-544: A return to the Soviet Union. In 1932 Emmy Noether and Emil Artin received the Ackermann–Teubner Memorial Award for their contributions to mathematics. The prize included a monetary reward of 500 ℛ︁ℳ︁ and was seen as a long-overdue official recognition of her considerable work in the field. Nevertheless, her colleagues expressed frustration at the fact that she was not elected to
3976-680: A series of other vowel shifts in the same region, known by linguists as the " Inland North ". The Inland North shares with the Eastern New England dialect (including Boston accents ) a backer tongue positioning of the GOOSE /u/ vowel (to [u] ) and the MOUTH /aʊ/ vowel (to [ɑʊ~äʊ] ) in comparison to the rest of the country. Ranging from northern New England across the Great Lakes to Minnesota, another Northern regional marker
4118-592: A soldier on the Eastern Front during World War II . Noether developed a close circle of mathematicians beyond just her doctoral students who shared Noether's approach to abstract algebra and contributed to the field's development, a group often referred to as the Noether school . An example of this is her close work with Wolfgang Krull , who greatly advanced commutative algebra with his Hauptidealsatz and his dimension theory for commutative rings. Another
4260-417: A subring shares the same multiplicative identity with the ambient ring R , if I were a subring, for every r ∈ R {\displaystyle r\in R} , we have r = r 1 ∈ I ; {\displaystyle r=r1\in I;} i.e., I = R {\displaystyle I=R} . The notion of an ideal does not involve associativity; thus, an ideal
4402-525: A survey, completed in 2003, polling English speakers across the United States about their specific everyday word choices, hoping to identify regionalisms. The study found that most Americans prefer the term sub for a long sandwich, soda (but pop in the Great Lakes region and generic coke in the South) for a sweet and bubbly soft drink , you or you guys for the plural of you (but y'all in
Emmy Noether - Misplaced Pages Continue
4544-847: A thesis on abelian groups and is remembered for his work in group theory , particularly Fitting's theorem and the Fitting lemma . He died at the age of 31 from a bone disease. Witt was initially supervised by Noether, but her position was revoked in April 1933 and he was assigned to Gustav Herglotz instead. He received his PhD in July 1933 with a thesis on the Riemann-Roch theorem and zeta-functions , and went on to make several contributions that now bear his name . Tsen, best remembered for proving Tsen's theorem , received his doctorate in December of
4686-453: A variation of American English in these islands. In 2021, about 245 million Americans, aged 5 or above, spoke English at home: a majority of the United States total population of roughly 330 million people. The United States has never had an official language at the federal level, but English is commonly used at the federal level and in states without an official language. 32 of the 50 states, in some cases as part of what has been called
4828-543: Is Gottfried Köthe , who contributed to the development of the theory of hypercomplex quantities using Noether and Krull's methods. In addition to her mathematical insight, Noether was respected for her consideration of others. Although she sometimes acted rudely toward those who disagreed with her, she nevertheless gained a reputation for constant helpfulness and patient guidance of new students. Her loyalty to mathematical precision caused one colleague to name her "a severe critic", but she combined this demand for accuracy with
4970-406: Is a surjective ring homomorphism that has the ideal as its kernel . Conversely, the kernel of a ring homomorphism is a two-sided ideal. Therefore, the two-sided ideals are exactly the kernels of ring homomorphisms. By convention, a ring has the multiplicative identity. But some authors do not require a ring to have the multiplicative identity; i.e., for them, a ring is a rng . For a rng R ,
5112-517: Is a generalization of an ideal, and the usual ideals are sometimes called integral ideals for clarity. Ernst Kummer invented the concept of ideal numbers to serve as the "missing" factors in number rings in which unique factorization fails; here the word "ideal" is in the sense of existing in imagination only, in analogy with "ideal" objects in geometry such as points at infinity. In 1876, Richard Dedekind replaced Kummer's undefined concept by concrete sets of numbers, sets that he called ideals, in
5254-415: Is a left ideal that is also a right ideal. If the ring is commutative , the three definitions are the same, and one talks simply of an ideal . In the non-commutative case, "ideal" is often used instead of "two-sided ideal". If I is a left, right or two-sided ideal, the relation x ∼ y {\displaystyle x\sim y} if and only if is an equivalence relation on R , and
5396-410: Is a maximal ideal. Conversely, if m {\displaystyle {\mathfrak {m}}} is a maximal ideal, then m {\displaystyle {\mathfrak {m}}} is the annihilator of the simple R -module R / m {\displaystyle R/{\mathfrak {m}}} . There is also another characterization (the proof is not hard): For
5538-654: Is also associated with the United States, perhaps mostly in the Midwest and the South. American accents that have not undergone the cot–caught merger (the lexical sets LOT and THOUGHT ) have instead retained a LOT – CLOTH split : a 17th-century distinction in which certain words (labeled as the CLOTH lexical set ) separated away from the LOT set. The split, which has now reversed in most British English, simultaneously shifts this relatively recent CLOTH set into
5680-820: Is also defined for non-associative rings (often without the multiplicative identity) such as a Lie algebra . (For the sake of brevity, some results are stated only for left ideals but are usually also true for right ideals with appropriate notation changes.) To simplify the description all rings are assumed to be commutative. The non-commutative case is discussed in detail in the respective articles. Ideals are important because they appear as kernels of ring homomorphisms and allow one to define factor rings . Different types of ideals are studied because they can be used to construct different types of factor rings. Two other important terms using "ideal" are not always ideals of their ring. See their respective articles for details: The sum and product of ideals are defined as follows. For
5822-664: Is also home to a creole language known commonly as Hawaiian Pidgin , and some Hawaii residents speak English with a Pidgin-influenced accent. American English also gave rise to some dialects outside the country, for example, Philippine English , beginning during the American occupation of the Philippines and subsequently the Insular Government of the Philippine Islands ; Thomasites first established
Emmy Noether - Misplaced Pages Continue
5964-532: Is an Artinian ring , then Jac ( R ) {\displaystyle \operatorname {Jac} (R)} is nilpotent and nil ( R ) = Jac ( R ) {\displaystyle \operatorname {nil} (R)=\operatorname {Jac} (R)} . (Proof: first note the DCC implies J n = J n + 1 {\displaystyle J^{n}=J^{n+1}} for some n . If (DCC)
6106-596: Is built-in to the definition of a Jacobson radical: if M is a module such that J M = M {\displaystyle JM=M} , then M does not admit a maximal submodule , since if there is a maximal submodule L ⊊ M {\displaystyle L\subsetneq M} , J ⋅ ( M / L ) = 0 {\displaystyle J\cdot (M/L)=0} and so M = J M ⊂ L ⊊ M {\displaystyle M=JM\subset L\subsetneq M} ,
6248-538: Is common in most American accents despite being now rare in England because, during the 17th-century British colonization, nearly all dialects of English were rhotic, and most North American English simply remained that way. The preservation of rhoticity in North America was also supported by continuing waves of rhotic-accented Scotch-Irish immigrants, most intensely during the 18th century (and moderately during
6390-420: Is often identified by Americans as a "country" accent, and is defined by the /aɪ/ vowel losing its gliding quality : [aː] , the initiation event for a complicated Southern vowel shift, including a " Southern drawl " that makes short front vowels into distinct-sounding gliding vowels . The fronting of the vowels of GOOSE , GOAT , MOUTH , and STRUT tends to also define Southern accents as well as
6532-411: Is the common language at home, in public, and in government. Ideal (ring theory) Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra In mathematics , and more specifically in ring theory , an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of
6674-565: Is the intersection of all primitive ideals. Equivalently, Indeed, if M {\displaystyle M} is a simple module and x is a nonzero element in M , then R x = M {\displaystyle Rx=M} and R / Ann ( M ) = R / Ann ( x ) ≃ M {\displaystyle R/\operatorname {Ann} (M)=R/\operatorname {Ann} (x)\simeq M} , meaning Ann ( M ) {\displaystyle \operatorname {Ann} (M)}
6816-640: Is the set of varieties of the English language native to the United States. English is the most widely spoken language in the United States ; the de facto common language used in government, education, and commerce; and an official language in 32 of the 50 U.S. states . Since the late 20th century, American English has become the most influential form of English worldwide. Varieties of American English include many patterns of pronunciation, vocabulary, grammar and particularly spelling that are unified nationwide but distinct from other English dialects around
6958-469: Is the variable fronting of /ɑ/ before /r/ , for example, appearing four times in the stereotypical Boston shibboleth Park the car in Harvard Yard . Several other phenomena serve to distinguish regional U.S. accents. Boston , Pittsburgh , Upper Midwestern , and Western U.S. accents have fully completed a merger of the LOT vowel with the THOUGHT vowel ( /ɑ/ and /ɔ/ , respectively):
7100-606: Is unknown. At about the same time, Noether's father retired and her brother joined the German Army to serve in World War I . She returned to Erlangen for several weeks, mostly to care for her aging father. During her first years teaching at Göttingen, she did not have an official position and was not paid. Her lectures often were advertised under Hilbert's name, and Noether would provide "assistance". Soon after arriving at Göttingen, she demonstrated her capabilities by proving
7242-482: The English-only movement , have adopted legislation granting official or co-official status to English. Typically only "English" is specified, not a particular variety like American English. (From 1923 to 1969, the state of Illinois recognized its official language as "American", meaning American English.) Puerto Rico is the largest example of a United States territory in which another language – Spanish –
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#17327730701007384-874: The German Reichskanzler in January ;1933, Nazi activity around the country increased dramatically. At the University of Göttingen, the German Student Association led the attack on the "un-German spirit" attributed to Jews and was aided by privatdozent and Noether's former student Werner Weber . Antisemitic attitudes created a climate hostile to Jewish professors. One young protester reportedly demanded: "Aryan students want Aryan mathematics and not Jewish mathematics." American English American English , sometimes called United States English or U.S. English ,
7526-621: The Göttingen Gesellschaft der Wissenschaften (academy of sciences) and was never promoted to the position of Ordentlicher Professor (full professor). Noether's colleagues celebrated her fiftieth birthday, in 1932, in typical mathematicians' style. Helmut Hasse dedicated an article to her in the Mathematische Annalen , wherein he confirmed her suspicion that some aspects of noncommutative algebra are simpler than those of commutative algebra , by proving
7668-528: The Hopkins–Levitzki theorem ) to ring theory . Other Noether Boys included Max Deuring , Hans Fitting , Ernst Witt , Chiungtze C. Tsen and Otto Schilling . Deuring, who had been considered the most promising of Noether's students, was awarded his doctorate in 1930. He worked in Hamburg, Marden and Göttingen and is known for his contributions to arithmetic geometry . Fitting graduated in 1931 with
7810-534: The Institute for Advanced Study in Princeton, New Jersey . Noether's mathematical work has been divided into three " epochs ". In the first (1908–1919), she made contributions to the theories of algebraic invariants and number fields . Her work on differential invariants in the calculus of variations , Noether's theorem , has been called "one of the most important mathematical theorems ever proved in guiding
7952-622: The Mid-Atlantic states (including a New York accent as well as a unique Philadelphia–Baltimore accent ), and the South . As of the 20th century, the middle and eastern Great Lakes area , Chicago being the largest city with these speakers, also ushered in certain unique features, including the fronting of the LOT /ɑ/ vowel in the mouth toward [a] and tensing of the TRAP /æ/ vowel wholesale to [eə] . These sound changes have triggered
8094-551: The Native American languages . Examples of such names are opossum , raccoon , squash , moose (from Algonquian ), wigwam , and moccasin . American English speakers have integrated traditionally non-English terms and expressions into the mainstream cultural lexicon; for instance, en masse , from French ; cookie , from Dutch ; kindergarten from German , and rodeo from Spanish . Landscape features are often loanwords from French or Spanish, and
8236-722: The francophile tastes of the 19th century Victorian era Britain (for example they preferred programme for program , manoeuvre for maneuver , cheque for check , etc.). AmE almost always uses -ize in words like realize . BrE prefers -ise , but also uses -ize on occasion (see: Oxford spelling ). There are a few differences in punctuation rules. British English is more tolerant of run-on sentences , called " comma splices " in American English, and American English prefers that periods and commas be placed inside closing quotation marks even in cases in which British rules would place them outside. American English also favors
8378-483: The inclusion of the ring of integers Z into the field of rationals Q ). The extension a e {\displaystyle {\mathfrak {a}}^{e}} of a {\displaystyle {\mathfrak {a}}} in B is defined to be the ideal in B generated by f ( a ) {\displaystyle f({\mathfrak {a}})} . Explicitly, If b {\displaystyle {\mathfrak {b}}}
8520-438: The integers , such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory , a normal subgroup can be used to construct
8662-412: The invariants of finite groups . This phase marked Noether's first exposure to abstract algebra , the field to which she would make groundbreaking contributions. In Erlangen, Noether advised two doctoral students: Hans Falckenberg and Fritz Seidelmann, who defended their theses in 1911 and 1916. Despite Noether's significant role, they were both officially under the supervision of her father. Following
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#17327730701008804-573: The theorem now known as Noether's theorem which shows that a conservation law is associated with any differentiable symmetry of a physical system . The paper, Invariante Variationsprobleme , was presented by a colleague, Felix Klein , on 26 July 1918 at a meeting of the Royal Society of Sciences at Göttingen. Noether presumably did not present it herself because she was not a member of the society. American physicists Leon M. Lederman and Christopher T. Hill argue in their book Symmetry and
8946-399: The "Noether Boys". In 1924, Dutch mathematician B. L. van der Waerden joined her circle and soon became the leading expositor of Noether's ideas; her work was the foundation for the second volume of his influential 1931 textbook, Moderne Algebra . By the time of her plenary address at the 1932 International Congress of Mathematicians in Zürich , her algebraic acumen was recognized around
9088-764: The 18th century; apartment , shanty in the 19th century; project, condominium , townhouse , mobile home in the 20th century; and parts thereof ( driveway , breezeway, backyard ) . Industry and material innovations from the 19th century onwards provide distinctive new words, phrases, and idioms through railroading (see further at rail terminology ) and transportation terminology, ranging from types of roads ( dirt roads , freeways ) to infrastructure ( parking lot , overpass , rest area ), to automotive terminology often now standard in English internationally. Already existing English words—such as store , shop , lumber —underwent shifts in meaning; others remained in
9230-580: The 1927–1928 academic year from the Rockefeller Foundation . In Göttingen, Noether supervised more than a dozen doctoral students, though most were together with Edmund Landau and others as she was not allowed to supervise dissertations on her own. Her first was Grete Hermann , who defended her dissertation in February ;1925. Although she is best remembered for her work on the foundations of quantum mechanics , her dissertation
9372-546: The 20th century. The use of English in the United States is a result of British colonization of the Americas . The first wave of English-speaking settlers arrived in North America during the early 17th century, followed by further migrations in the 18th and 19th centuries. During the 17th and 18th centuries, dialects from many different regions of England and the British Isles existed in every American colony, allowing
9514-696: The Beautiful Universe that Noether's theorem is "certainly one of the most important mathematical theorems ever proved in guiding the development of modern physics , possibly on a par with the Pythagorean theorem ". When World War I ended, the German Revolution of 1918–1919 brought a significant change in social attitudes, including more rights for women. In 1919 the University of Göttingen allowed Noether to proceed with her habilitation (eligibility for tenure). Her oral examination
9656-627: The British form is a back-formation , such as AmE burglarize and BrE burgle (from burglar ). However, while individuals usually use one or the other, both forms will be widely understood and mostly used alongside each other within the two systems. While written American English is largely standardized across the country and spoken American English dialects are highly mutually intelligible, there are still several recognizable regional and ethnic accents and lexical distinctions. The regional sounds of present-day American English are reportedly engaged in
9798-413: The East Coast (perhaps in imitation of 19th-century London speech), even the East Coast has gradually begun to restore rhoticity, due to it becoming nationally prestigious in the 20th century. The pronunciation of ⟨r⟩ is a postalveolar approximant [ ɹ̠ ] or retroflex approximant [ ɻ ] , but a unique "bunched tongue" variant of the approximant r sound
9940-571: The Inland North. Rather than one particular accent, General American is best defined as an umbrella covering an American accent that does not incorporate features associated with some particular region, ethnicity, or socioeconomic group. Typical General American features include rhoticity , the father–bother merger , Mary–marry–merry merger , pre-nasal "short a " tensing , and other particular vowel sounds . General American features are embraced most by Americans who are highly educated or in
10082-573: The South), sneakers for athletic shoes (but often tennis shoes outside the Northeast), and shopping cart for a cart used for carrying supermarket goods. American English and British English (BrE) often differ at the levels of phonology, phonetics, vocabulary, and, to a much lesser extent, grammar and orthography. The first large American dictionary, An American Dictionary of the English Language , known as Webster's Dictionary ,
10224-528: The U.S. Several verbs ending in -ize are of U.S. origin; for example, fetishize, prioritize, burglarize, accessorize, weatherize , etc.; and so are some back-formations (locate, fine-tune, curate, donate, emote, upholster and enthuse). Among syntactic constructions that arose are outside of, headed for, meet up with, back of, etc. Americanisms formed by alteration of some existing words include notably pesky, phony, rambunctious, buddy, sundae , skeeter, sashay and kitty-corner. Adjectives that arose in
10366-570: The U.S. are for instance foothill , landslide (in all senses), backdrop , teenager , brainstorm , bandwagon , hitchhike , smalltime, and a huge number of others. Other compound words have been founded based on industrialization and the wave of the automobile: five-passenger car, four-door sedan, two-door sedan, and station-wagon (called an estate car in British English). Some are euphemistic ( human resources , affirmative action , correctional facility ). Many compound nouns have
10508-676: The U.S. are, for example, lengthy, bossy, cute and cutesy, punk (in all senses), sticky (of the weather), through (as in "finished"), and many colloquial forms such as peppy or wacky . A number of words and meanings that originated in Middle English or Early Modern English and that have been in everyday use in the United States have since disappeared in most varieties of British English; some of these have cognates in Lowland Scots . Terms such as fall ("autumn"), faucet ("tap"), diaper ("nappy"; itself unused in
10650-530: The U.S. while changing in Britain. Science, urbanization, and democracy have been important factors in bringing about changes in the written and spoken language of the United States. From the world of business and finance came new terms ( merger , downsize , bottom line ), from sports and gambling terminology came, specific jargon aside, common everyday American idioms, including many idioms related to baseball . The names of some American inventions remained largely confined to North America ( elevator [except in
10792-427: The U.S.), candy ("sweets"), skillet , eyeglasses , and obligate are often regarded as Americanisms. Fall for example came to denote the season in 16th century England, a contraction of Middle English expressions like "fall of the leaf" and "fall of the year." Gotten ( past participle of get ) is often considered to be largely an Americanism. Other words and meanings were brought back to Britain from
10934-597: The U.S., especially in the second half of the 20th century; these include hire ("to employ"), I guess (famously criticized by H. W. Fowler ), baggage , hit (a place), and the adverbs overly and presently ("currently"). Some of these, for example, monkey wrench and wastebasket , originated in 19th century Britain. The adjectives mad meaning "angry", smart meaning "intelligent", and sick meaning "ill" are also more frequent in American (and Irish) English than British English. Linguist Bert Vaux created
11076-538: The United States and the United Kingdom suggest that, while spoken American English deviated away from period British English in many ways, it is conservative in a few other ways, preserving certain features 21st-century British English has since lost. Full rhoticity (or "R-fulness") is typical of American accents, pronouncing the phoneme /r/ (corresponding to the letter ⟨r⟩ ) in all environments, including in syllable-final position or before
11218-449: The United States. Noether's other students were Wilhelm Dörnte, who received his doctorate in 1927 with a thesis on groups, Werner Vorbeck, who did so in 1935 with a thesis on splitting fields , and Wolfgang Wichmann, who did so 1936 with a thesis on p-adic theory . There is no information about the first two, but it is known that Wichmann supported a student initiative that unsuccessfully attempted to revoke Noether's dismissal and died as
11360-450: The University of Göttingen. He immediately began working with Noether, who provided invaluable methods of abstract conceptualization. Van der Waerden later said that her originality was "absolute beyond comparison". After returning to Amsterdam, he wrote Moderne Algebra , a central two-volume text in the field; its second volume, published in 1931, borrowed heavily from Noether's work. Although Noether did not seek recognition, he included as
11502-605: The West and Midwest, and New York Latino English , spoken in the New York metropolitan area . Additionally, ethnic varieties such as Yeshiva English and " Yinglish " are spoken by some American Orthodox Jews , Cajun Vernacular English by some Cajuns in southern Louisiana , and Pennsylvania Dutch English by some Pennsylvania Dutch people. American Indian Englishes have been documented among diverse Indian tribes. The island state of Hawaii , though primarily English-speaking,
11644-555: The accents spoken in the " Midland ": a vast band of the country that constitutes an intermediate dialect region between the traditional North and South. Western U.S. accents mostly fall under the General American spectrum. Below, ten major American English accents are defined by their particular combinations of certain vowel sounds: In 2010, William Labov noted that Great Lakes, Philadelphia, Pittsburgh, and West Coast accents have undergone "vigorous new sound changes" since
11786-505: The aeronautical sense ], gasoline ) as did certain automotive terms ( truck , trunk ). New foreign loanwords came with 19th and early 20th century European immigration to the U.S.; notably, from Yiddish ( chutzpah , schmooze, bupkis, glitch ) and German ( hamburger , wiener ). A large number of English colloquialisms from various periods are American in origin; some have lost their American flavor (from OK and cool to nerd and 24/7 ), while others have not ( have
11928-447: The basis for several important textbooks, such as those of van der Waerden and Deuring. Noether transmitted an infectious mathematical enthusiasm to her most dedicated students, who relished their lively conversations with her. Several of her colleagues attended her lectures and she sometimes allowed others (including her students) to receive credit for her ideas, resulting in much of her work appearing in papers not under her name. Noether
12070-527: The completion of his doctorate, Falckenberg spent time in Braunschweig and Königsberg before becoming a professor at the University of Giessen while Seidelmann became a professor in Munich . In the spring of 1915, Noether was invited to return to the University of Göttingen by David Hilbert and Felix Klein . Their effort to recruit her was initially blocked by the philologists and historians among
12212-406: The country), though the vowel-consonant cluster found in "bird", "work", "hurt", "learn", etc. usually retains its r pronunciation, even in these non-rhotic American accents. Non-rhoticity among such speakers is presumed to have arisen from their upper classes' close historical contact with England, imitating London's r -dropping, a feature that has continued to gain prestige throughout England from
12354-667: The development of Galois theory . Although politics was not central to her life, Noether took a keen interest in political matters and, according to Alexandrov, showed considerable support for the Russian Revolution . She was especially happy to see Soviet advances in the fields of science and mathematics, which she considered indicative of new opportunities made possible by the Bolshevik project. This attitude caused her problems in Germany, culminating in her eviction from
12496-583: The development of modern physics". In the second epoch (1920–1926), she began work that "changed the face of [abstract] algebra". In her classic 1921 paper Idealtheorie in Ringbereichen ( Theory of Ideals in Ring Domains ), Noether developed the theory of ideals in commutative rings into a tool with wide-ranging applications. She made elegant use of the ascending chain condition , and objects satisfying it are named Noetherian in her honor. In
12638-614: The diverse regional dialects of British English) became common after the mid-18th century, while at the same time speakers' identification with this new variety increased. Since the 18th century, American English has developed into some new varieties, including regional dialects that retain minor influences from waves of immigrant speakers of diverse languages, primarily European languages. Some racial and regional variation in American English reflects these groups' geographic settlement, their de jure or de facto segregation, and patterns in their resettlement. This can be seen, for example, in
12780-742: The double quotation mark ("like this") over the single ('as here'). Vocabulary differences vary by region. For example, autumn is used more commonly in the United Kingdom, whereas fall is more common in American English. Some other differences include: aerial (United Kingdom) vs. antenna, biscuit (United Kingdom) vs. cookie/cracker, car park (United Kingdom) vs. parking lot, caravan (United Kingdom) vs. trailer, city centre (United Kingdom) vs. downtown, flat (United Kingdom) vs. apartment, fringe (United Kingdom) vs. bangs, and holiday (United Kingdom) vs. vacation. AmE sometimes favors words that are morphologically more complex, whereas BrE uses clipped forms, such as AmE transportation and BrE transport or where
12922-500: The elements of the ring. For instance, the prime ideals of a ring are analogous to prime numbers , and the Chinese remainder theorem can be generalized to ideals. There is a version of unique prime factorization for the ideals of a Dedekind domain (a type of ring important in number theory ). The related, but distinct, concept of an ideal in order theory is derived from the notion of ideal in ring theory. A fractional ideal
13064-409: The first of several in 1923. Between 1926 and 1930, Alexandrov regularly lectured at the university, and he and Noether became good friends. He dubbed her der Noether , using der as an epithet rather than as the masculine German article. She tried to arrange for him to obtain a position at Göttingen as a regular professor, but was able only to help him secure a scholarship to Princeton University for
13206-517: The following two centuries) when this ethnic group eventually made up one-seventh of the colonial population. Scotch-Irish settlers spread from Delaware and Pennsylvania throughout the larger Mid-Atlantic region, the inland regions of both the South and North, and throughout the West: American dialect areas that were all uninfluenced by upper-class non-rhoticity and that consequently have remained consistently rhotic. While non-rhoticity spread on
13348-499: The form ab with a in a {\displaystyle {\mathfrak {a}}} and b in b {\displaystyle {\mathfrak {b}}} . Note a + b {\displaystyle {\mathfrak {a}}+{\mathfrak {b}}} is the smallest left (resp. right) ideal containing both a {\displaystyle {\mathfrak {a}}} and b {\displaystyle {\mathfrak {b}}} (or
13490-589: The former in 1911. According to her colleague Hermann Weyl and her biographer Auguste Dick , Fischer was an important influence on Noether, in particular by introducing her to the work of David Hilbert . Noether and Fischer shared lively enjoyment of mathematics and would often discuss lectures long after they were over; Noether is known to have sent postcards to Fischer continuing her train of mathematical thoughts. From 1913 to 1916, Noether published several papers extending and applying Hilbert's methods to mathematical objects such as fields of rational functions and
13632-644: The handkerchief from her blouse and ignored the increasing disarray of her hair during a lecture. Two female students once approached her during a break in a two-hour class to express their concern, but they were unable to break through the energetic mathematical discussion she was having with other students. Noether did not follow a lesson plan for her lectures. She spoke quickly and her lectures were considered difficult to follow by many, including Carl Ludwig Siegel and Paul Dubreil . Students who disliked her style often felt alienated. "Outsiders" who occasionally visited Noether's lectures usually spent only half an hour in
13774-944: The hospital , BrE to hospital ; contrast, however, AmE actress Elizabeth Taylor , BrE the actress Elizabeth Taylor ). Often, these differences are a matter of relative preferences rather than absolute rules; and most are not stable since the two varieties are constantly influencing each other, and American English is not a standardized set of dialects. Differences in orthography are also minor. The main differences are that American English usually uses spellings such as flavor for British flavour , fiber for fibre , defense for defence , analyze for analyse , license for licence , catalog for catalogue and traveling for travelling . Noah Webster popularized such spellings in America, but he did not invent most of them. Rather, "he chose already existing options on such grounds as simplicity, analogy or etymology." Other differences are due to
13916-674: The influence of 18th-century Protestant Ulster Scots immigrants (known in the U.S. as the Scotch-Irish ) in Appalachia developing Appalachian English and the 20th-century Great Migration bringing African-American Vernacular English to the Great Lakes urban centers. Any phonologically unmarked North American accent falls under an umbrella known as General American. This section mostly refers to such General American features. Studies on historical usage of English in both
14058-473: The intersection of ideals is again an ideal; with these two operations as join and meet, the set of all ideals of a given ring forms a complete modular lattice . The lattice is not, in general, a distributive lattice . The three operations of intersection, sum (or join), and product make the set of ideals of a commutative ring into a quantale . If a , b {\displaystyle {\mathfrak {a}},{\mathfrak {b}}} are ideals of
14200-438: The late 18th century onwards, but which has conversely lost prestige in the U.S. since at least the early 20th century. Non-rhoticity makes a word like car sound like cah or source like sauce . New York City and Southern accents are the most prominent regional accents of the country, as well as the most stigmatized and socially disfavored. Southern speech, strongest in southern Appalachia and certain areas of Texas,
14342-566: The left by elements of R {\displaystyle R} "; that is, I {\displaystyle I} is a left ideal if it satisfies the following two conditions: In other words, a left ideal is a left submodule of R , considered as a left module over itself. A right ideal is defined similarly, with the condition r x ∈ I {\displaystyle rx\in I} replaced by x r ∈ I {\displaystyle xr\in I} . A two-sided ideal
14484-487: The mathematics department at the University of Göttingen , a world-renowned center of mathematical research. The philosophical faculty objected, however, and she spent four years lecturing under Hilbert's name. Her habilitation was approved in 1919, allowing her to obtain the rank of Privatdozent . Noether remained a leading member of the Göttingen mathematics department until 1933; her students were sometimes called
14626-427: The mid-nineteenth century onwards, so they "are now more different from each other than they were 50 or 100 years ago", while other accents, like of New York City and Boston, have remained stable in that same time-frame. However, a General American sound system also has some debated degree of influence nationwide, for example, gradually beginning to oust the regional accent in urban areas of the South and at least some in
14768-418: The most distinctive innovations of twentieth century mathematics, is largely due to her – in published papers, in lectures, and in personal influence on her contemporaries. Noether's work in algebra began in 1920 when, in collaboration with her protégé Werner Schmeidler, she published a paper about the theory of ideals in which they defined left and right ideals in a ring . The following year she published
14910-617: The most formal contexts, and regional accents with the most General American native features include North Midland, Western New England, and Western accents. Although no longer region-specific, African-American Vernacular English , which remains the native variety of most working- and middle-class African Americans , has a close relationship to Southern dialects and has greatly influenced everyday speech of many Americans, including hip hop culture . Hispanic and Latino Americans have also developed native-speaker varieties of English. The best-studied Latino Englishes are Chicano English , spoken in
15052-519: The most important woman in the history of mathematics . As one of the leading mathematicians of her time, she developed theories of rings , fields , and algebras . In physics, Noether's theorem explains the connection between symmetry and conservation laws . Noether was born to a Jewish family in the Franconian town of Erlangen ; her father was the mathematician Max Noether . She originally planned to teach French and English after passing
15194-517: The paper Idealtheorie in Ringbereichen , analyzing ascending chain conditions with regards to (mathematical) ideals , in which she proved the Lasker–Noether theorem in its full generality. Noted algebraist Irving Kaplansky called this work "revolutionary". The publication gave rise to the term Noetherian for objects which satisfy the ascending chain condition. In 1924, a young Dutch mathematician, Bartel Leendert van der Waerden , arrived at
15336-402: The past forms of a few verbs (for example, AmE/BrE: learned / learnt , burned / burnt , snuck/sneaked , dove/dived ) although the purportedly "British" forms can occasionally be seen in American English writing as well; different prepositions and adverbs in certain contexts (for example, AmE in school, BrE at school ); and whether or not a definite article is used, in very few cases (AmE to
15478-405: The philosophical faculty, who insisted that women should not become privatdozenten . In a joint department meeting on the matter, one faculty member protested: "What will our soldiers think when they return to the university and find that they are required to learn at the feet of a woman?" Hilbert, who believed Noether's qualifications were the only important issue and that the sex of the candidate
15620-489: The required examinations but instead studied mathematics at the University of Erlangen , where her father lectured. After completing her doctorate in 1907 under the supervision of Paul Gordan , she worked at the Mathematical Institute of Erlangen without pay for seven years. At the time, women were largely excluded from academic positions. In 1915, she was invited by David Hilbert and Felix Klein to join
15762-402: The room before leaving in frustration or confusion. A regular student said of one such instance: "The enemy has been defeated; he has cleared out." She used her lectures as a spontaneous discussion time with her students, to think through and clarify important problems in mathematics. Some of her most important results were developed in these lectures, and the lecture notes of her students formed
15904-408: The same year. He returned to China in 1935 and started teaching at National Chekiang University , but died only five years later. Schilling also began studying under Noether but was forced to find a new advisor due to Noether's emigration. Under Helmut Hasse , he completed his PhD in 1934 at the University of Marburg . He later worked as a post doc at Trinity College, Cambridge , before moving to
16046-463: The set of equivalence classes forms a left, right or bi module denoted R / I {\displaystyle R/I} and called the quotient of R by I . (It is an instance of a congruence relation and is a generalization of modular arithmetic .) If the ideal I is two-sided, R / I {\displaystyle R/I} is a ring, and the function that associates to each element of R its equivalence class
16188-475: The spring of 1900, she took the examination for teachers of these languages and received an overall score of sehr gut (very good). Her performance qualified her to teach languages at schools reserved for girls, but she chose instead to continue her studies at the University of Erlangen , at which her father was a professor. This was an unconventional decision; two years earlier, the Academic Senate of
16330-475: The third edition of Dirichlet 's book Vorlesungen über Zahlentheorie , to which Dedekind had added many supplements. Later the notion was extended beyond number rings to the setting of polynomial rings and other commutative rings by David Hilbert and especially Emmy Noether . Given a ring R , a left ideal is a subset I of R that is a subgroup of the additive group of R {\displaystyle R} that "absorbs multiplication from
16472-443: The third epoch (1927–1935), she published works on noncommutative algebras and hypercomplex numbers and united the representation theory of groups with the theory of modules and ideals. In addition to her own publications, Noether was generous with her ideas and is credited with several lines of research published by other mathematicians, even in fields far removed from her main work, such as algebraic topology . Emmy Noether
16614-412: The title of nicht beamteter ausserordentlicher Professor (an untenured professor with limited internal administrative rights and functions). This was an unpaid "extraordinary" professorship , not the higher "ordinary" professorship, which was a civil-service position. Although it recognized the importance of her work, the position still provided no salary. Noether was not paid for her lectures until she
16756-570: The traditional standard accent of (southern) England, Received Pronunciation (RP), has evolved a trap–bath split . Moreover, American accents preserve /h/ at the start of syllables, while perhaps a majority of the regional dialects of England participate in /h/ dropping , particularly in informal contexts. However, General American is also innovative in a number of its own ways: The process of coining new lexical items started as soon as English-speaking British-American colonists began borrowing names for unfamiliar flora, fauna, and topography from
16898-508: The union a ∪ b {\displaystyle {\mathfrak {a}}\cup {\mathfrak {b}}} ), while the product a b {\displaystyle {\mathfrak {a}}{\mathfrak {b}}} is contained in the intersection of a {\displaystyle {\mathfrak {a}}} and b {\displaystyle {\mathfrak {b}}} . The distributive law holds for two-sided ideals
17040-563: The union of their generators. In the last three we observe that products and intersections agree whenever the two ideals intersect in the zero ideal. These computations can be checked using Macaulay2 . Ideals appear naturally in the study of modules, especially in the form of a radical. Let R be a commutative ring. By definition, a primitive ideal of R is the annihilator of a (nonzero) simple R -module . The Jacobson radical J = Jac ( R ) {\displaystyle J=\operatorname {Jac} (R)} of R
17182-411: The university had declared that allowing mixed-sex education would "overthrow all academic order". One of just two women in a university of 986 students, Noether was allowed only to audit classes rather than participate fully, and she required the permission of individual professors whose lectures she wished to attend. Despite these obstacles, on 14 July 1903, she passed the graduation exam at
17324-505: The university in October 1904, declaring her intention to focus solely on mathematics. She was one of six women in her year (two auditors) and the only woman in her chosen school. Under the supervision of Paul Gordan , she wrote her dissertation, Über die Bildung des Formensystems der ternären biquadratischen Form ( On Complete Systems of Invariants for Ternary Biquadratic Forms ), in 1907, graduating summa cum laude later that year. Gordan
17466-417: The verb-and-preposition combination: stopover, lineup, tryout, spin-off, shootout , holdup, hideout, comeback, makeover , and many more. Some prepositional and phrasal verbs are in fact of American origin ( win out, hold up, back up/off/down/out, face up to and many others). Noun endings such as -ee (retiree), -ery (bakery), -ster (gangster) and -cian (beautician) are also particularly productive in
17608-417: The woods, and lectured at a local coffee house. Later, after Nazi Germany dismissed her from teaching, she invited students into her home to discuss their plans for the future and mathematical concepts. Noether's frugal lifestyle was at first due to her being denied pay for her work. However, even after the university began paying her a small salary in 1923, she continued to live a simple and modest life. She
17750-775: The word corn , used in England to refer to wheat (or any cereal), came to denote the maize plant, the most important crop in the U.S. Most Mexican Spanish contributions came after the War of 1812 , with the opening of the West, like ranch (now a common house style ). Due to Mexican culinary influence, many Spanish words are incorporated in general use when talking about certain popular dishes: cilantro (instead of coriander), queso, tacos, quesadillas, enchiladas, tostadas, fajitas, burritos, and guacamole. These words usually lack an English equivalent and are found in popular restaurants. New forms of dwelling created new terms ( lot , waterfront) and types of homes like log cabin , adobe in
17892-609: The world. Any American or Canadian accent perceived as lacking noticeably local, ethnic, or cultural markers is known in linguistics as General American ; it covers a fairly uniform accent continuum native to certain regions of the U.S. but especially associated with broadcast mass media and highly educated speech. However, historical and present linguistic evidence does not support the notion of there being one single mainstream American accent . The sound of American English continues to evolve, with some local accents disappearing, but several larger regional accents having emerged in
18034-464: The world. The following year, Germany's Nazi government dismissed Jews from university positions , and Noether moved to the United States to take up a position at Bryn Mawr College in Pennsylvania . There, she taught graduate and post-doctoral women including Marie Johanna Weiss , Ruth Stauffer, Grace Shover Quinn, and Olga Taussky-Todd . At the same time, she lectured and performed research at
18176-463: Was near-sighted and talked with a minor lisp during her childhood. A family friend recounted a story years later about young Noether quickly solving a brain teaser at a children's party, showing logical acumen at an early age. She was taught to cook and clean, as were most girls of the time, and took piano lessons. She pursued none of these activities with passion, although she loved to dance. She had three younger brothers. The eldest, Alfred Noether,
18318-418: Was a member of the "computational" school of invariant researchers, and Noether's thesis ended with a list of over 300 explicitly worked-out invariants. This approach to invariants was later superseded by the more abstract and general approach pioneered by Hilbert. Although it had been well received, Noether later described her thesis and some subsequent similar papers she produced as "crap". All of her later work
18460-402: Was appointed to the special position of Lehrbeauftragte für Algebra a year later. Although Noether's theorem had a significant effect upon classical and quantum mechanics, among mathematicians she is best remembered for her contributions to abstract algebra . In his introduction to Noether's Collected Papers , Nathan Jacobson wrote that The development of abstract algebra, which is one of
18602-435: Was attended by 800 people, including Noether's colleagues Hermann Weyl , Edmund Landau , and Wolfgang Krull . There were 420 official participants and twenty-one plenary addresses presented. Apparently, Noether's prominent speaking position was a recognition of the importance of her contributions to mathematics. The 1932 congress is sometimes described as the high point of her career. When Adolf Hitler became
18744-514: Was born in 1883 and was awarded a doctorate in chemistry from Erlangen in 1909, but died nine years later. Fritz Noether was born in 1884, studied in Munich and made contributions to applied mathematics . He was executed in the Soviet Union in 1941. The youngest, Gustav Robert Noether, was born in 1889. Very little is known about his life; he suffered from chronic illness and died in 1928. Noether showed early proficiency in French and English. In
18886-434: Was born on 23 March 1882. She was the first of four children of mathematician Max Noether and Ida Amalia Kaufmann, both from Jewish merchant families. Her first name was "Amalie", but she began using her middle name at a young age and she invariably used the name "Emmy Noether" in her adult life and her publications. In her youth, Noether did not stand out academically although she was known for being clever and friendly. She
19028-485: Was considered an important contribution to ideal theory . Hermann later spoke reverently of her "dissertation-mother". Around the same time, Heinrich Grell and Rudolf Hölzer wrote their dissertations under Noether, though the latter died of tuberculosis shortly before his defense. Grell defended his thesis in 1926 and went on to work at the University of Jena and the University of Halle , before losing his teaching license in 1935 due to accusations of homosexual acts. He
19170-543: Was held in late May, and she successfully delivered her habilitation lecture in June 1919. Noether became a privatdozent , and she delivered that fall semester the first lectures listed under her own name. She was still not paid for her work. Three years later, she received a letter from Otto Boelitz [ de ] , the Prussian Minister for Science, Art, and Public Education, in which he conferred on her
19312-438: Was in a completely different field. From 1908 to 1915, Noether taught at Erlangen's Mathematical Institute without pay, occasionally substituting for her father, Max Noether , when he was too ill to lecture. In 1910 and 1911, she published an extension of her thesis work from three variables to n variables. Gordan retired in 1910, and Noether taught under his successors, Erhard Schmidt and Ernst Fischer , who took over from
19454-705: Was irrelevant, objected with indignation and scolded those protesting her habilitation. Although his exact words have not been preserved, his objection is often said to have included the remark that the university was "not a bathhouse." According to Pavel Alexandrov 's recollection, faculty members' opposition to Noether was based not just in sexism, but also in their objections to her social-democratic political beliefs and Jewish ancestry. Noether left for Göttingen in late April; two weeks later her mother died suddenly in Erlangen. She had previously received medical care for an eye condition, but its nature and impact on her death
19596-611: Was later reinstated and became a professor at Humboldt University in 1948. Noether then supervised Werner Weber and Jakob Levitzki , who both defended their theses in 1929. Weber, who was considered only a modest mathematician, would later take part in driving Jewish mathematicians out of Göttingen. Levitzki worked first at Yale University and then at the Hebrew University of Jerusalem in Palestine, making significant contributions (in particular Levitzky's theorem and
19738-487: Was not a Bolshevist but was not afraid to be called one." Noether planned to return to Moscow, an effort for which she received support from Alexandrov. After she left Germany in 1933, he tried to help her gain a chair at Moscow State University through the Soviet Education Ministry . Although this effort proved unsuccessful, they corresponded frequently during the 1930s, and in 1935 she made plans for
19880-570: Was paid more generously later in her life, but saved half of her salary to bequeath to her nephew, Gottfried E. Noether . Biographers suggest that she was mostly unconcerned about appearance and manners, focusing on her studies. Olga Taussky-Todd , a distinguished algebraist taught by Noether, described a luncheon during which Noether, wholly engrossed in a discussion of mathematics, "gesticulated wildly" as she ate and "spilled her food constantly and wiped it off from her dress, completely unperturbed". Appearance-conscious students cringed as she retrieved
20022-447: Was recorded as having given at least five semester-long courses at Göttingen: In the winter of 1928–1929, Noether accepted an invitation to Moscow State University , where she continued working with P.S. Alexandrov . In addition to carrying on with her research, she taught classes in abstract algebra and algebraic geometry . She worked with the topologists Lev Pontryagin and Nikolai Chebotaryov , who later praised her contributions to
20164-469: Was written by Noah Webster in 1828, codifying several of these spellings. Differences in grammar are relatively minor, and do not normally affect mutual intelligibility; these include: typically a lack of differentiation between adjectives and adverbs, employing the equivalent adjectives as adverbs he ran quick / he ran quickly ; different use of some auxiliary verbs ; formal (rather than notional) agreement with collective nouns ; different preferences for
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