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68-615: Going Up may refer to: Going up and going down , terms in commutative algebra which refer to certain properties of chains of prime ideals in integral extensions Going Up (musical) , a musical comedy that opened in New York in 1917 and in London in 1918 Going Up (film) , a 1923 film starring Douglas MacLean "Going Up" (TV episode) , an episode of PBS's POV series Going Up (2007 film) , starring Nandita Chandra "Going Up",

136-411: A {\displaystyle a} divides a product b c {\displaystyle bc} , a {\displaystyle a} divides b {\displaystyle b} or c {\displaystyle c} . In a domain, being prime implies being irreducible. The converse is true in a unique factorization domain, but false in general. The definition of ideals

204-406: A ⋅ b {\displaystyle a\cdot b} . To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition; i.e., a ⋅ ( b + c ) = ( a ⋅ b ) + (

272-402: A ⋅ c ) {\displaystyle a\cdot \left(b+c\right)=\left(a\cdot b\right)+\left(a\cdot c\right)} . The identity elements for addition and multiplication are denoted 0 {\displaystyle 0} and 1 {\displaystyle 1} , respectively. If the multiplication is commutative, i.e. a ⋅ b = b ⋅

340-406: A + I ) ( b + I ) = a b + I {\displaystyle \left(a+I\right)\left(b+I\right)=ab+I} . For example, the ring Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } (also denoted Z n {\displaystyle \mathbb {Z} _{n}} ), where n {\displaystyle n}

408-498: A + b ) n = ∑ k = 0 n ( n k ) a k b n − k {\displaystyle (a+b)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}a^{k}b^{n-k}} which is valid for any two elements a and b in any commutative ring R is understood in this sense by interpreting the binomial coefficients as elements of R using this map. Given two R -algebras S and T , their tensor product

476-439: A , {\displaystyle a\cdot b=b\cdot a,} then the ring R {\displaystyle R} is called commutative . In the remainder of this article, all rings will be commutative, unless explicitly stated otherwise. An important example, and in some sense crucial, is the ring of integers Z {\displaystyle \mathbb {Z} } with the two operations of addition and multiplication. As

544-425: A basis is called a free module , and a submodule of a free module needs not to be free. A module of finite type is a module that has a finite spanning set. Modules of finite type play a fundamental role in the theory of commutative rings, similar to the role of the finite-dimensional vector spaces in linear algebra . In particular, Noetherian rings (see also § Noetherian rings , below) can be defined as

612-441: A chain such that q i {\displaystyle {\mathfrak {q}}_{i}} lies over p i {\displaystyle {\mathfrak {p}}_{i}} for each 1 ≤  i  ≤  n . In ( Kaplansky 1970 ) it is shown that if an extension A  ⊆  B satisfies the going-up property, then it also satisfies the lying-over property. The ring extension A  ⊆  B

680-399: A chain such that q i {\displaystyle {\mathfrak {q}}_{i}} lies over p i {\displaystyle {\mathfrak {p}}_{i}} for each 1 ≤  i  ≤  n . There is a generalization of the ring extension case with ring morphisms. Let f  : A  →  B be a (unital) ring homomorphism so that B

748-439: A commutative ring. The same is true for differentiable or holomorphic functions , when the two concepts are defined, such as for V {\displaystyle V} a complex manifold . In contrast to fields, where every nonzero element is multiplicatively invertible, the concept of divisibility for rings is richer. An element a {\displaystyle a} of ring R {\displaystyle R}

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816-455: A domain is called irreducible if the only way of expressing it as a product a = b c , {\displaystyle a=bc,} is by either b {\displaystyle b} or c {\displaystyle c} being a unit. An example, important in field theory , are irreducible polynomials , i.e., irreducible elements in k [ X ] {\displaystyle k\left[X\right]} , for

884-423: A field k {\displaystyle k} . The fact that Z {\displaystyle \mathbb {Z} } is a UFD can be stated more elementarily by saying that any natural number can be uniquely decomposed as product of powers of prime numbers. It is also known as the fundamental theorem of arithmetic . An element a {\displaystyle a} is a prime element if whenever

952-402: A field k can be axiomatized by four properties: The dimension is defined, for any ring R , as the supremum of lengths n of chains of prime ideals For example, a field is zero-dimensional, since the only prime ideal is the zero ideal. The integers are one-dimensional, since chains are of the form (0) ⊊ ( p ), where p is a prime number . For non-Noetherian rings, and also non-local rings,

1020-464: A prime p {\displaystyle {\mathfrak {p}}} in A , then q {\displaystyle {\mathfrak {q}}}  ⊈  q ′ {\displaystyle {\mathfrak {q}}'} and q ′ {\displaystyle {\mathfrak {q}}'}  ⊈  q {\displaystyle {\mathfrak {q}}} . The ring extension A  ⊆  B

1088-578: A prime ideal is principal, it is equivalently generated by a prime element. However, in rings such as Z [ − 5 ] , {\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right],} prime ideals need not be principal. This limits the usage of prime elements in ring theory. A cornerstone of algebraic number theory is, however, the fact that in any Dedekind ring (which includes Z [ − 5 ] {\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right]} and more generally

1156-430: A principal ideal domain, the properties of individual elements are strongly tied to the properties of the ring as a whole. For example, any principal ideal domain R {\displaystyle R} is a unique factorization domain (UFD) which means that any element is a product of irreducible elements, in a (up to reordering of factors) unique way. Here, an element a {\displaystyle a} in

1224-534: A product: 6 = 2 ⋅ 3 = ( 1 + − 5 ) ( 1 − − 5 ) . {\displaystyle 6=2\cdot 3=\left(1+{\sqrt {-5}}\right)\left(1-{\sqrt {-5}}\right).} Prime ideals, as opposed to prime elements, provide a way to circumvent this problem. A prime ideal is a proper (i.e., strictly contained in R {\displaystyle R} ) ideal p {\displaystyle p} such that, whenever

1292-438: A ring extension A  ⊆  B : There is another sufficient condition for the going-down property: Proof : Let p 1  ⊆  p 2 be prime ideals of A and let q 2 be a prime ideal of B such that q 2  ∩  A  =  p 2 . We wish to prove that there is a prime ideal q 1 of B contained in q 2 such that q 1  ∩  A  =  p 1 . Since A  ⊆  B

1360-421: A ring homomorphism is thus a map that is compatible with the structure of the algebraic objects in question. In such a situation S is also called an R -algebra, by understanding that s in S may be multiplied by some r of R , by setting The kernel and image of f are defined by ker( f ) = { r ∈ R , f ( r ) = 0} and im( f ) = f ( R ) = { f ( r ), r ∈ R } . The kernel is an ideal of R , and

1428-421: A single element r {\displaystyle r} , the ideal generated by F {\displaystyle F} consists of the multiples of r {\displaystyle r} , i.e., the elements of the form r s {\displaystyle rs} for arbitrary elements s {\displaystyle s} . Such an ideal is called a principal ideal . If every ideal

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1496-623: A song by Echo & the Bunnymen from their 1980 album Crocodiles "Going Up", a common announcement played in elevators Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Going Up . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Going_Up&oldid=1049855756 " Category : Disambiguation pages Hidden categories: Short description

1564-413: Is surjective and there exists a prime ideal of B q 2 that contracts to the prime ideal p 1 A p 2 of A p 2 . The contraction of this prime ideal of B q 2 to B is a prime ideal q 1 of B contained in q 2 that contracts to p 1 . The proof is complete.  Q.E.D. Commutative ring In mathematics , a commutative ring is a ring in which

1632-415: Is a set R {\displaystyle R} equipped with two binary operations , i.e. operations combining any two elements of the ring to a third. They are called addition and multiplication and commonly denoted by " + {\displaystyle +} " and " ⋅ {\displaystyle \cdot } "; e.g. a + b {\displaystyle a+b} and

1700-514: Is a field, called the quotient field of R {\displaystyle R} . Many of the following notions also exist for not necessarily commutative rings, but the definitions and properties are usually more complicated. For example, all ideals in a commutative ring are automatically two-sided , which simplifies the situation considerably. For a ring R {\displaystyle R} , an R {\displaystyle R} - module M {\displaystyle M}

1768-695: Is a field. Except for the zero ring , any ring (with identity) possesses at least one maximal ideal; this follows from Zorn's lemma . A ring is called Noetherian (in honor of Emmy Noether , who developed this concept) if every ascending chain of ideals 0 ⊆ I 0 ⊆ I 1 ⊆ ⋯ ⊆ I n ⊆ I n + 1 … {\displaystyle 0\subseteq I_{0}\subseteq I_{1}\subseteq \dots \subseteq I_{n}\subseteq I_{n+1}\dots } becomes stationary, i.e. becomes constant beyond some index n {\displaystyle n} . Equivalently, any ideal

1836-399: Is a flat extension of rings, it follows that A p 2  ⊆  B q 2 is a flat extension of rings. In fact, A p 2  ⊆  B q 2 is a faithfully flat extension of rings since the inclusion map A p 2  →  B q 2 is a local homomorphism. Therefore, the induced map on spectra Spec( B q 2 ) → Spec( A p 2 )

1904-465: Is a principal ideal, R {\displaystyle R} is called a principal ideal ring ; two important cases are Z {\displaystyle \mathbb {Z} } and k [ X ] {\displaystyle k\left[X\right]} , the polynomial ring over a field k {\displaystyle k} . These two are in addition domains, so they are called principal ideal domains . Unlike for general rings, for

1972-402: Is a product of pairwise distinct prime numbers . Commutative rings, together with ring homomorphisms, form a category . The ring Z is the initial object in this category, which means that for any commutative ring R , there is a unique ring homomorphism Z → R . By means of this map, an integer n can be regarded as an element of R . For example, the binomial formula (

2040-483: Is a ring extension of f ( A ). Then f is said to satisfy the going-up property if the going-up property holds for f ( A ) in  B . Similarly, if B is a ring extension of f ( A ), then f is said to satisfy the going-down property if the going-down property holds for f ( A ) in B . In the case of ordinary ring extensions such as A  ⊆  B , the inclusion map is the pertinent map. The usual statements of going-up and going-down theorems refer to

2108-421: Is an integer, is the ring of integers modulo n {\displaystyle n} . It is the basis of modular arithmetic . An ideal is proper if it is strictly smaller than the whole ring. An ideal that is not strictly contained in any proper ideal is called maximal . An ideal m {\displaystyle m} is maximal if and only if R / m {\displaystyle R/m}

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2176-415: Is any ring element. Interpreting f {\displaystyle f} as a function that takes the value f mod p (i.e., the image of f in the residue field R / p ), this subset is the locus where f is non-zero. The spectrum also makes precise the intuition that localisation and factor rings are complementary: the natural maps R → R f and R → R / fR correspond, after endowing

2244-433: Is automatically a prime ideal of A ) then we say that p {\displaystyle {\mathfrak {p}}} lies under q {\displaystyle {\mathfrak {q}}} and that q {\displaystyle {\mathfrak {q}}} lies over p {\displaystyle {\mathfrak {p}}} . In general, a ring extension A  ⊆  B of commutative rings

2312-445: Is called a unit if it possesses a multiplicative inverse. Another particular type of element is the zero divisors , i.e. an element a {\displaystyle a} such that there exists a non-zero element b {\displaystyle b} of the ring such that a b = 0 {\displaystyle ab=0} . If R {\displaystyle R} possesses no non-zero zero divisors, it

2380-415: Is called an integral domain (or domain). An element a {\displaystyle a} satisfying a n = 0 {\displaystyle a^{n}=0} for some positive integer n {\displaystyle n} is called nilpotent . The localization of a ring is a process in which some elements are rendered invertible, i.e. multiplicative inverses are added to

2448-521: Is different from Wikidata All article disambiguation pages All disambiguation pages Going up and going down The phrase going up refers to the case when a chain can be extended by "upward inclusion ", while going down refers to the case when a chain can be extended by "downward inclusion". The major results are the Cohen–Seidenberg theorems , which were proved by Irvin S. Cohen and Abraham Seidenberg . These are known as

2516-853: Is generated by finitely many elements, or, yet equivalent, submodules of finitely generated modules are finitely generated. Being Noetherian is a highly important finiteness condition, and the condition is preserved under many operations that occur frequently in geometry. For example, if R {\displaystyle R} is Noetherian, then so is the polynomial ring R [ X 1 , X 2 , … , X n ] {\displaystyle R\left[X_{1},X_{2},\dots ,X_{n}\right]} (by Hilbert's basis theorem ), any localization S − 1 R {\displaystyle S^{-1}R} , and also any factor ring R / I {\displaystyle R/I} . Any non-Noetherian ring R {\displaystyle R}

2584-425: Is important enough to have its own notation: R p {\displaystyle R_{p}} . This ring has only one maximal ideal, namely p R p {\displaystyle pR_{p}} . Such rings are called local . The spectrum of a ring R {\displaystyle R} , denoted by Spec   R {\displaystyle {\text{Spec}}\ R} ,

2652-416: Is in bijection with the set Thus, maximal ideals reflect the geometric properties of solution sets of polynomials, which is an initial motivation for the study of commutative rings. However, the consideration of non-maximal ideals as part of the geometric properties of a ring is useful for several reasons. For example, the minimal prime ideals (i.e., the ones not strictly containing smaller ones) correspond to

2720-423: Is invertible; i.e., has a multiplicative inverse b {\displaystyle b} such that a ⋅ b = 1 {\displaystyle a\cdot b=1} . Therefore, by definition, any field is a commutative ring. The rational , real and complex numbers form fields. If R {\displaystyle R} is a given commutative ring, then the set of all polynomials in

2788-464: Is like what a vector space is to a field. That is, elements in a module can be added; they can be multiplied by elements of R {\displaystyle R} subject to the same axioms as for a vector space. The study of modules is significantly more involved than the one of vector spaces , since there are modules that do not have any basis , that is, do not contain a spanning set whose elements are linearly independents . A module that has

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2856-407: Is prime, or equivalently that a ring has no zero-divisors can be very difficult. Yet another way of expressing the same is to say that the complement R ∖ p {\displaystyle R\setminus p} is multiplicatively closed. The localisation ( R ∖ p ) − 1 R {\displaystyle \left(R\setminus p\right)^{-1}R}

2924-424: Is said to satisfy the going-down property if whenever is a chain of prime ideals of A and is a chain of prime ideals of B with m < n and such that q i {\displaystyle {\mathfrak {q}}_{i}} lies over p i {\displaystyle {\mathfrak {p}}_{i}} for 1 ≤  i  ≤  m , then the latter chain can be extended to

2992-422: Is said to satisfy the going-up property if whenever is a chain of prime ideals of A and is a chain of prime ideals of B with m < n and such that q i {\displaystyle {\mathfrak {q}}_{i}} lies over p i {\displaystyle {\mathfrak {p}}_{i}} for 1 ≤  i  ≤  m , then the latter chain can be extended to

3060-542: Is said to satisfy the lying over property if every prime ideal p {\displaystyle {\mathfrak {p}}} of A lies under some prime ideal q {\displaystyle {\mathfrak {q}}} of  B . The extension A  ⊆  B is said to satisfy the incomparability property if whenever q {\displaystyle {\mathfrak {q}}} and q ′ {\displaystyle {\mathfrak {q}}'} are distinct primes of B lying over

3128-514: Is some index set), the ideal generated by F {\displaystyle F} is the smallest ideal that contains F {\displaystyle F} . Equivalently, it is given by finite linear combinations r 1 f 1 + r 2 f 2 + ⋯ + r n f n . {\displaystyle r_{1}f_{1}+r_{2}f_{2}+\dots +r_{n}f_{n}.} If F {\displaystyle F} consists of

3196-509: Is such that "dividing" I {\displaystyle I} "out" gives another ring, the factor ring R / I {\displaystyle R/I} : it is the set of cosets of I {\displaystyle I} together with the operations ( a + I ) + ( b + I ) = ( a + b ) + I {\displaystyle \left(a+I\right)+\left(b+I\right)=\left(a+b\right)+I} and (

3264-585: Is the union of its Noetherian subrings. This fact, known as Noetherian approximation , allows the extension of certain theorems to non-Noetherian rings. A ring is called Artinian (after Emil Artin ), if every descending chain of ideals R ⊇ I 0 ⊇ I 1 ⊇ ⋯ ⊇ I n ⊇ I n + 1 … {\displaystyle R\supseteq I_{0}\supseteq I_{1}\supseteq \dots \supseteq I_{n}\supseteq I_{n+1}\dots } becomes stationary eventually. Despite

3332-417: Is the common basis of commutative algebra and algebraic geometry . Algebraic geometry proceeds by endowing Spec R with a sheaf O {\displaystyle {\mathcal {O}}} (an entity that collects functions defined locally, i.e. on varying open subsets). The datum of the space and the sheaf is called an affine scheme . Given an affine scheme, the underlying ring R can be recovered as

3400-588: Is the set of all prime ideals of R {\displaystyle R} . It is equipped with a topology, the Zariski topology , which reflects the algebraic properties of R {\displaystyle R} : a basis of open subsets is given by D ( f ) = { p ∈ Spec   R , f ∉ p } , {\displaystyle D\left(f\right)=\left\{p\in {\text{Spec}}\ R,f\not \in p\right\},} where f {\displaystyle f}

3468-694: The Hopkins–Levitzki theorem , every Artinian ring is Noetherian. More precisely, Artinian rings can be characterized as the Noetherian rings whose Krull dimension is zero. As was mentioned above, Z {\displaystyle \mathbb {Z} } is a unique factorization domain . This is not true for more general rings, as algebraists realized in the 19th century. For example, in Z [ − 5 ] {\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right]} there are two genuinely distinct ways of writing 6 as

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3536-429: The global sections of O {\displaystyle {\mathcal {O}}} . Moreover, this one-to-one correspondence between rings and affine schemes is also compatible with ring homomorphisms: any f  : R → S gives rise to a continuous map in the opposite direction The resulting equivalence of the two said categories aptly reflects algebraic properties of rings in a geometrical manner. Similar to

3604-700: The going-up and going-down theorems . Let A  ⊆  B be an extension of commutative rings . The going-up and going-down theorems give sufficient conditions for a chain of prime ideals in B , each member of which lies over members of a longer chain of prime ideals in A , to be able to be extended to the length of the chain of prime ideals in A . First, we fix some terminology. If p {\displaystyle {\mathfrak {p}}} and q {\displaystyle {\mathfrak {q}}} are prime ideals of A and B , respectively, such that (note that q ∩ A {\displaystyle {\mathfrak {q}}\cap A}

3672-461: The irreducible components of Spec R . For a Noetherian ring R , Spec R has only finitely many irreducible components. This is a geometric restatement of primary decomposition , according to which any ideal can be decomposed as a product of finitely many primary ideals . This fact is the ultimate generalization of the decomposition into prime ideals in Dedekind rings. The notion of a spectrum

3740-411: The ring of integers in a number field ) any ideal (such as the one generated by 6) decomposes uniquely as a product of prime ideals. Any maximal ideal is a prime ideal or, more briefly, is prime. Moreover, an ideal I {\displaystyle I} is prime if and only if the factor ring R / I {\displaystyle R/I} is an integral domain. Proving that an ideal

3808-591: The cancellation familiar from rational numbers. Indeed, in this language Q {\displaystyle \mathbb {Q} } is the localization of Z {\displaystyle \mathbb {Z} } at all nonzero integers. This construction works for any integral domain R {\displaystyle R} instead of Z {\displaystyle \mathbb {Z} } . The localization ( R ∖ { 0 } ) − 1 R {\displaystyle \left(R\setminus \left\{0\right\}\right)^{-1}R}

3876-494: The dimension may be infinite, but Noetherian local rings have finite dimension. Among the four axioms above, the first two are elementary consequences of the definition, whereas the remaining two hinge on important facts in commutative algebra , the going-up theorem and Krull's principal ideal theorem . A ring homomorphism or, more colloquially, simply a map , is a map f  : R → S such that These conditions ensure f (0) = 0 . Similarly as for other algebraic structures,

3944-437: The fact that manifolds are locally given by open subsets of R , affine schemes are local models for schemes , which are the object of study in algebraic geometry. Therefore, several notions concerning commutative rings stem from geometric intuition. The Krull dimension (or dimension) dim R of a ring R measures the "size" of a ring by, roughly speaking, counting independent elements in R . The dimension of algebras over

4012-705: The ideals of a ring is of particular importance, but often one proceeds by studying modules in general. Any ring has two ideals, namely the zero ideal { 0 } {\displaystyle \left\{0\right\}} and R {\displaystyle R} , the whole ring. These two ideals are the only ones precisely if R {\displaystyle R} is a field. Given any subset F = { f j } j ∈ J {\displaystyle F=\left\{f_{j}\right\}_{j\in J}} of R {\displaystyle R} (where J {\displaystyle J}

4080-480: The image is a subring of S . A ring homomorphism is called an isomorphism if it is bijective. An example of a ring isomorphism, known as the Chinese remainder theorem , is Z / n = ⨁ i = 0 k Z / p i , {\displaystyle \mathbf {Z} /n=\bigoplus _{i=0}^{k}\mathbf {Z} /p_{i},} where n = p 1 p 2 ... p k

4148-458: The multiplication of integers is a commutative operation, this is a commutative ring. It is usually denoted Z {\displaystyle \mathbb {Z} } as an abbreviation of the German word Zahlen (numbers). A field is a commutative ring where 0 ≠ 1 {\displaystyle 0\not =1} and every non-zero element a {\displaystyle a}

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4216-562: The multiplication operation is commutative . The study of commutative rings is called commutative algebra . Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. Ring homomorphisms Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra A ring

4284-412: The product a b {\displaystyle ab} of any two ring elements a {\displaystyle a} and b {\displaystyle b} is in p , {\displaystyle p,} at least one of the two elements is already in p . {\displaystyle p.} (The opposite conclusion holds for any ideal, by definition.) Thus, if

4352-735: The ring. Concretely, if S {\displaystyle S} is a multiplicatively closed subset of R {\displaystyle R} (i.e. whenever s , t ∈ S {\displaystyle s,t\in S} then so is s t {\displaystyle st} ) then the localization of R {\displaystyle R} at S {\displaystyle S} , or ring of fractions with denominators in S {\displaystyle S} , usually denoted S − 1 R {\displaystyle S^{-1}R} consists of symbols subject to certain rules that mimic

4420-917: The rings such that every submodule of a module of finite type is also of finite type. Ideals of a ring R {\displaystyle R} are the submodules of R {\displaystyle R} , i.e., the modules contained in R {\displaystyle R} . In more detail, an ideal I {\displaystyle I} is a non-empty subset of R {\displaystyle R} such that for all r {\displaystyle r} in R {\displaystyle R} , i {\displaystyle i} and j {\displaystyle j} in I {\displaystyle I} , both r i {\displaystyle ri} and i + j {\displaystyle i+j} are in I {\displaystyle I} . For various applications, understanding

4488-537: The spectra of the rings in question with their Zariski topology, to complementary open and closed immersions respectively. Even for basic rings, such as illustrated for R = Z at the right, the Zariski topology is quite different from the one on the set of real numbers. The spectrum contains the set of maximal ideals, which is occasionally denoted mSpec ( R ). For an algebraically closed field k , mSpec (k[ T 1 , ..., T n ] / ( f 1 , ..., f m ))

4556-524: The two conditions appearing symmetric, Noetherian rings are much more general than Artinian rings. For example, Z {\displaystyle \mathbb {Z} } is Noetherian, since every ideal can be generated by one element, but is not Artinian, as the chain Z ⊋ 2 Z ⊋ 4 Z ⊋ 8 Z … {\displaystyle \mathbb {Z} \supsetneq 2\mathbb {Z} \supsetneq 4\mathbb {Z} \supsetneq 8\mathbb {Z} \dots } shows. In fact, by

4624-579: The variable X {\displaystyle X} whose coefficients are in R {\displaystyle R} forms the polynomial ring , denoted R [ X ] {\displaystyle R\left[X\right]} . The same holds true for several variables. If V {\displaystyle V} is some topological space , for example a subset of some R n {\displaystyle \mathbb {R} ^{n}} , real- or complex-valued continuous functions on V {\displaystyle V} form

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