[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/graded-ring-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/graded-ring-wikipedia\/","headline":"Graded ring – Wikipedia","name":"Graded ring – Wikipedia","description":"before-content-x4 In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is","datePublished":"2022-05-05","dateModified":"2022-05-05","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/db421291be9d0103404ced7495b363437b67b6b1","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/db421291be9d0103404ced7495b363437b67b6b1","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/graded-ring-wikipedia\/","about":["Wiki"],"wordCount":14018,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups Ri{displaystyle R_{i}}       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4 such that RiRj\u2286Ri+j{displaystyle R_{i}R_{j}subseteq R_{i+j}}. The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading.A graded module is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Z{displaystyle mathbb {Z} }-algebra.The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to non-associative algebras as well; e.g., one can consider a graded Lie algebra.Table of Contents       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4First properties[edit]Basic examples[edit]Graded module[edit]Invariants of graded modules[edit]Graded algebra[edit]G-graded rings and algebras[edit]Anticommutativity[edit]Examples[edit]Graded monoid[edit]Power series indexed by a graded monoid[edit]Example[edit]See also[edit]References[edit]First properties[edit]Generally, the index set of a graded ring is assumed to be the set of nonnegative integers, unless otherwise explicitly specified. This is the case in this article.A graded ring is a ring that is decomposed into a direct sumR=\u2a01n=0\u221eRn=R0\u2295R1\u2295R2\u2295\u22ef{displaystyle R=bigoplus _{n=0}^{infty }R_{n}=R_{0}oplus R_{1}oplus R_{2}oplus cdots }ofadditive groups, such thatRmRn\u2286Rm+n{displaystyle R_{m}R_{n}subseteq R_{m+n}}for all nonnegative integers m{displaystyle m} and n{displaystyle n}.A nonzero element of Rn{displaystyle R_{n}} is said to be homogeneous of degree n{displaystyle n}. By definition of a direct sum, every nonzero element a{displaystyle a} of R{displaystyle R} can be uniquely written as a sum a=a0+a1+\u22ef+an{displaystyle a=a_{0}+a_{1}+cdots +a_{n}} where each ai{displaystyle a_{i}} is either 0 or homogeneous of degree i{displaystyle i}. The nonzero ai{displaystyle a_{i}} are the homogeneous components of\u00a0a{displaystyle a}.Some basic properties are:An ideal I\u2286R{displaystyle Isubseteq R} is homogeneous, if for every a\u2208I{displaystyle ain I}, the homogeneous components of a{displaystyle a} also belong to I.{displaystyle I.} (Equivalently, if it is a graded submodule of R{displaystyle R}; see \u00a7\u00a0Graded module.) The intersection of a homogeneous ideal I{displaystyle I} with Rn{displaystyle R_{n}} is an R0{displaystyle R_{0}}-submodule of Rn{displaystyle R_{n}} called the homogeneous part of degree n{displaystyle n} of I{displaystyle I}. A homogeneous ideal is the direct sum of its homogeneous parts.If I{displaystyle I} is a two-sided homogeneous ideal in R{displaystyle R}, then R\/I{displaystyle R\/I} is also a graded ring, decomposed asR\/I=\u2a01n=0\u221eRn\/In,{displaystyle R\/I=bigoplus _{n=0}^{infty }R_{n}\/I_{n},}where In{displaystyle I_{n}} is the homogeneous part of degree n{displaystyle n} of I{displaystyle I}.Basic examples[edit]Any (non-graded) ring R can be given a gradation by letting R0=R{displaystyle R_{0}=R}, and Ri=0{displaystyle R_{i}=0} for i \u2260 0.  This is called the trivial gradation on\u00a0R.The polynomial ring R=k[t1,\u2026,tn]{displaystyle R=k[t_{1},ldots ,t_{n}]} is graded by degree: it is a direct sum of Ri{displaystyle R_{i}} consisting of homogeneous polynomials of degree i.Let S be the set of all nonzero homogeneous elements in a graded integral domain R. Then the localization of R with respect to S is a Z{displaystyle mathbb {Z} }-graded ring.If I is an ideal in a commutative ring R, then \u2a01n=0\u221eIn\/In+1{displaystyle bigoplus _{n=0}^{infty }I^{n}\/I^{n+1}} is a graded ring called the associated graded ring of R along I; geometrically, it is the coordinate ring of the normal cone along the subvariety defined by I.Let X be a topological space, H\u200ai(X; R) the ith cohomology group with coefficients in a ring R. Then H\u2009*(X; R), the cohomology ring of X with coefficients in R, is a graded ring whose underlying group is \u2a01i=0\u221eHi(X;R){displaystyle bigoplus _{i=0}^{infty }H^{i}(X;R)} with the multiplicative structure given by the cup product.Graded module[edit]The corresponding idea in module theory is that of a graded module, namely a left module M over a graded ring R such that alsoM=\u2a01i\u2208NMi,{displaystyle M=bigoplus _{iin mathbb {N} }M_{i},}andRiMj\u2286Mi+j.{displaystyle R_{i}M_{j}subseteq M_{i+j}.}Example: a graded vector space is an example of a graded module over a field (with the field having trivial grading).Example: a graded ring is a graded module over itself. An ideal in a graded ring is homogeneous if and only if it is a graded submodule. The annihilator of a graded module is a homogeneous ideal.Example: Given an ideal I in a commutative ring R and an R-module M, the direct sum \u2a01n=0\u221eInM\/In+1M{displaystyle bigoplus _{n=0}^{infty }I^{n}M\/I^{n+1}M} is a graded module over the associated graded ring \u2a010\u221eIn\/In+1{displaystyle bigoplus _{0}^{infty }I^{n}\/I^{n+1}}.A morphism f:N\u2192M{displaystyle f:Nto M} between graded modules, called a graded morphism, is a morphism of underlying modules that respects grading; i.e., f(Ni)\u2286Mi{displaystyle f(N_{i})subseteq M_{i}}. A graded submodule is a submodule that is a graded module in own right and such that the set-theoretic inclusion is a morphism of graded modules. Explicitly, a graded module N is a graded submodule of M if and only if it is a submodule of M and satisfies Ni=N\u2229Mi{displaystyle N_{i}=Ncap M_{i}}. The kernel and the image of a morphism of graded modules are graded submodules.Remark: To give a graded morphism from a graded ring to another graded ring with the image lying in the center is the same as to give the structure of a graded algebra to the latter ring.Given a graded module M{displaystyle M}, the \u2113{displaystyle ell }-twist of M{displaystyle M} is a graded module defined by M(\u2113)n=Mn+\u2113{displaystyle M(ell )_{n}=M_{n+ell }}. (cf. Serre’s twisting sheaf in algebraic geometry.)Let M and N be graded modules. If f:M\u2192N{displaystyle fcolon Mto N} is a morphism of modules, then f is said to have degree d if f(Mn)\u2286Nn+d{displaystyle f(M_{n})subseteq N_{n+d}}. An exterior derivative of differential forms in differential geometry is an example of such a morphism having degree 1.Invariants of graded modules[edit]Given a graded module M over a commutative graded ring R, one can associate the formal power series P(M,t)\u2208Z[[t]]{displaystyle P(M,t)in mathbb {Z} [![t]!]}:P(M,t)=\u2211\u2113(Mn)tn{displaystyle P(M,t)=sum ell (M_{n})t^{n}}(assuming \u2113(Mn){displaystyle ell (M_{n})} are finite.) It is called the Hilbert\u2013Poincar\u00e9 series of M.A graded module is said to be finitely generated if the underlying module is finitely generated. The generators may be taken to be homogeneous (by replacing the generators by their homogeneous parts.)Suppose R is a polynomial ring k[x0,\u2026,xn]{displaystyle k[x_{0},dots ,x_{n}]}, k a field, and M a finitely generated graded module over it. Then the function n\u21a6dimk\u2061Mn{displaystyle nmapsto dim _{k}M_{n}} is called the Hilbert function of M. The function coincides with the integer-valued polynomial for large n called the Hilbert polynomial of M.Graded algebra[edit]An algebra A over a ring R is a graded algebra if it is graded as a ring.In the usual case where the ring R is not graded (in particular if R is a field), it is given the trivial grading (every element of R is of degree 0). Thus, R\u2286A0{displaystyle Rsubseteq A_{0}} and the graded pieces Ai{displaystyle A_{i}} are R-modules.In the case where the ring R is also a graded ring, then one requires thatRiAj\u2286Ai+j{displaystyle R_{i}A_{j}subseteq A_{i+j}}In other words, we require A to be a graded left module over R.Examples of graded algebras are common in mathematics:Graded algebras are much used in commutative algebra and algebraic geometry, homological algebra, and algebraic topology. One example is the close relationship between homogeneous polynomials and projective varieties (cf. Homogeneous coordinate ring.)G-graded rings and algebras[edit]The above definitions have been generalized to rings graded using any monoid G as an index set.  A G-graded ring R is a ring with a direct sum decompositionR=\u2a01i\u2208GRi{displaystyle R=bigoplus _{iin G}R_{i}}such thatRiRj\u2286Ri\u22c5j.{displaystyle R_{i}R_{j}subseteq R_{icdot j}.}Elements of R that lie inside Ri{displaystyle R_{i}} for some i\u2208G{displaystyle iin G} are said to be homogeneous of grade i.The previously defined notion of “graded ring” now becomes the same thing as an N{displaystyle mathbb {N} }-graded ring, where N{displaystyle mathbb {N} } is the monoid of natural numbers under addition. The definitions for graded modules and algebras can also be extended this way replacing the indexing set N{displaystyle mathbb {N} } with any monoid G.Remarks:If we do not require that the ring have an identity element, semigroups may replace monoids.Examples:Anticommutativity[edit]Some graded rings (or algebras) are endowed with an anticommutative structure.  This notion requires a homomorphism of the monoid of the gradation into the additive monoid of Z\/2Z{displaystyle mathbb {Z} \/2mathbb {Z} }, the field with two elements.  Specifically, a signed monoid consists of a pair (\u0393,\u03b5){displaystyle (Gamma ,varepsilon )} where \u0393{displaystyle Gamma } is a monoid and \u03b5:\u0393\u2192Z\/2Z{displaystyle varepsilon colon Gamma to mathbb {Z} \/2mathbb {Z} } is a homomorphism of additive monoids.  An anticommutative \u0393{displaystyle Gamma }-graded ring is a ring A graded with respect to \u0393 such that:xy=(\u22121)\u03b5(deg\u2061x)\u03b5(deg\u2061y)yx,{displaystyle xy=(-1)^{varepsilon (deg x)varepsilon (deg y)}yx,}for all homogeneous elements x and y.Examples[edit]Graded monoid[edit]Intuitively, a graded monoid is the subset of a graded ring, \u2a01n\u2208N0Rn{displaystyle bigoplus _{nin mathbb {N} _{0}}R_{n}}, generated by the Rn{displaystyle R_{n}}‘s, without using the additive part. That is, the set of elements of the graded monoid is \u22c3n\u2208N0Rn{displaystyle bigcup _{nin mathbb {N} _{0}}R_{n}}.Formally, a graded monoid[1] is a monoid (M,\u22c5){displaystyle (M,cdot )}, with a gradation function \u03d5:M\u2192N0{displaystyle phi :Mto mathbb {N} _{0}} such that \u03d5(m\u22c5m\u2032)=\u03d5(m)+\u03d5(m\u2032){displaystyle phi (mcdot m’)=phi (m)+phi (m’)}. Note that the gradation of 1M{displaystyle 1_{M}} is necessarily 0. Some authors request furthermore that \u03d5(m)\u22600{displaystyle phi (m)neq 0}when m is not the identity.Assuming the gradations of non-identity elements are non-zero, the number of elements of gradation n is at most gn{displaystyle g^{n}} where g is the cardinality of a generating set G of the monoid. Therefore the number of elements of gradation n or less is at most n+1{displaystyle n+1} (for g=1{displaystyle g=1}) or gn+1\u22121g\u22121{displaystyle {frac {g^{n+1}-1}{g-1}}} else. Indeed, each such element is the product of at most n elements of G, and only gn+1\u22121g\u22121{displaystyle {frac {g^{n+1}-1}{g-1}}} such products exist.  Similarly, the identity element can not be written as the product of two non-identity elements. That is, there is no unit divisor in such a graded monoid.Power series indexed by a graded monoid[edit]This notions allows to extends the notion of power series ring. Instead of having the indexing family being N{displaystyle mathbb {N} }, the indexing family could be any graded monoid, assuming that the number of elements of degree n is finite, for each integer n.More formally, let (K,+K,\u00d7K){displaystyle (K,+_{K},times _{K})} be an arbitrary semiring and (R,\u22c5,\u03d5){displaystyle (R,cdot ,phi )} a graded monoid. Then K\u27e8\u27e8R\u27e9\u27e9{displaystyle Klangle langle Rrangle rangle } denotes the semiring of power series with coefficients in K indexed by R. Its elements are functions from R to K. The sum of two elements s,s\u2032\u2208K\u27e8\u27e8R\u27e9\u27e9{displaystyle s,s’in Klangle langle Rrangle rangle } is defined pointwise, it is the function sending m\u2208R{displaystyle min R} to s(m)+Ks\u2032(m){displaystyle s(m)+_{K}s'(m)}, and the product is the function sending m\u2208R{displaystyle min R} to the infinite sum \u2211p,q\u2208Rp\u22c5q=ms(p)\u00d7Ks\u2032(q){displaystyle sum _{p,qin R atop pcdot q=m}s(p)times _{K}s'(q)}. This sum is correctly defined (i.e., finite) because, for each m, there are only a finite number of pairs (p, q) such that pq = m.Example[edit]In formal language theory, given an alphabet A, the free monoid of words over A can be considered as a graded monoid, where the gradation of a word is its length.See also[edit]References[edit]^ Sakarovitch, Jacques (2009). “Part II: The power of algebra”. Elements of automata theory. Translated by Thomas, Reuben. Cambridge University Press. p.\u00a0384. ISBN\u00a0978-0-521-84425-3. Zbl\u00a01188.68177.Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol.\u00a0211 (Revised third\u00a0ed.), New York: Springer-Verlag, ISBN\u00a0978-0-387-95385-4, MR\u00a01878556.Bourbaki, N. (1974). “Ch. 1\u20133, 3 \u00a73”. Algebra I. ISBN\u00a0978-3-540-64243-5.Steenbrink, J. (1977). “Intersection form for quasi-homogeneous singularities” (PDF). Compositio Mathematica. 34 (2): 211\u2013223 See p. 211. ISSN\u00a00010-437X.Matsumura, H. (1989). “5 Dimension theory \u00a7S3 Graded rings, the Hilbert function and the Samuel function”. Commutative Ring Theory. Cambridge Studies in Advanced Mathematics. Vol.\u00a08. Translated by Reid, M. (2nd\u00a0ed.). Cambridge University Press. ISBN\u00a0978-1-107-71712-1.     (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/graded-ring-wikipedia\/#breadcrumbitem","name":"Graded ring – Wikipedia"}}]}]