Editing Graded ring

{{Short description|Type of algebraic structure}}
{{Algebraic structures |Algebra}}
In [[mathematics]], in particular [[abstract algebra]], a '''graded ring''' is a [[ring (mathematics)|ring]] such that the underlying [[additive group]] is a [[direct sum of abelian groups]] <math>R_i</math> such that {{tmath|1= R_i R_j \subseteq R_{i+j} }}. The [[index set]] is usually the set of nonnegative [[integer]]s 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 space]]s. A graded module that is also a graded ring is called a '''graded algebra'''. A graded ring could also be viewed as a graded {{tmath|1= \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 algebra]]s as well; e.g., one can consider a [[graded Lie algebra]].

== First properties ==
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 (mathematics)|ring]] that is decomposed into a [[direct sum]]
: <math>R = \bigoplus_{n=0}^\infty R_n = R_0 \oplus R_1 \oplus R_2 \oplus \cdots</math> 
of 
[[additive group]]s, such that 
: <math>R_mR_n \subseteq R_{m+n}</math>
for all nonnegative integers <math>m</math> and {{tmath|1= n }}.

A nonzero element of <math>R_n</math> is said to be ''homogeneous'' of ''degree'' {{tmath|1= n }}. By definition of a direct sum, every nonzero element <math>a</math> of <math>R</math> can be uniquely written as a sum <math>a=a_0+a_1+\cdots +a_n</math> where each <math>a_i</math> is either 0 or homogeneous of degree {{tmath|1= i }}. The nonzero <math>a_i</math> are the ''homogeneous components'' of&nbsp;{{tmath|1= a }}.

Some basic properties are:
* <math>R_0</math> is a [[subring]] of {{tmath|1= R }}; in particular, the multiplicative identity <math>1</math> is a homogeneous element of degree zero.
* For any <math>n</math>, <math>R_n</math> is a two-sided {{tmath|1= R_0 }}-[[module (mathematics)|module]], and the direct sum decomposition is a direct sum of {{tmath|1= R_0 }}-modules.
* <math>R</math> is an [[associative algebra|associative {{tmath|1= R_0 }}-algebra]].

An [[ideal (ring theory)|ideal]] <math>I\subseteq R</math> is ''homogeneous'', if for every {{tmath|1= a \in I }}, the homogeneous components of <math>a</math> also belong to {{tmath|1= I }}. (Equivalently, if it is a graded submodule of {{tmath|1= R }}; see {{section link||Graded module}}.) The [[intersection (set theory)|intersection]] of a homogeneous ideal <math>I</math> with <math>R_n</math> is an {{tmath|1= R_0 }}-[[submodule]] of <math>R_n</math> called the ''homogeneous part'' of degree <math>n</math> of {{tmath|1= I }}. A homogeneous ideal is the direct sum of its homogeneous parts.

If <math>I</math> is a two-sided homogeneous ideal in {{tmath|1= R }}, then <math>R/I</math> is also a graded ring, decomposed as
: <math>R/I = \bigoplus_{n=0}^\infty R_n/I_n,</math>
where <math>I_n</math> is the homogeneous part of degree <math>n</math> of {{tmath|1= I }}.

== Basic examples ==
* Any (non-graded) ring ''R'' can be given a gradation by letting {{tmath|1= R_0=R }}, and <math>R_i=0</math> for ''i'' ≠ 0.  This is called the '''trivial gradation''' on&nbsp;''R''.
* The [[polynomial ring]] <math>R = k[t_1, \ldots, t_n]</math> is graded by [[Degree of a polynomial|degree]]: it is a direct sum of <math>R_i</math> consisting of [[homogeneous polynomial]]s of degree ''i''.
* Let ''S'' be the set of all nonzero homogeneous elements in a graded [[integral domain]] ''R''. Then the [[localization of a ring|localization]] of ''R'' with respect to ''S'' is a <math>\Z</math>-graded ring.
* If ''I'' is an ideal in a [[commutative ring]] ''R'', then <math display=inline>\bigoplus_{n=0}^{\infty} I^n/I^{n+1}</math> 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 [[algebraic variety|subvariety]] defined by ''I''.
* Let ''X'' be a [[topological space]], ''H''<sup>{{hairsp}}''i''</sup>(''X''; ''R'') the ''i''th [[cohomology group]] with coefficients in a ring ''R''. Then ''H''<sup>&thinsp;*</sup>(''X''; ''R''), the [[cohomology ring]] of ''X'' with coefficients in ''R'', is a graded ring whose underlying [[abelian group|group]] is <math display=inline>\bigoplus_{i = 0}^\infty H^i(X; R)</math> with the multiplicative structure given by the [[cup product]].

== Graded module ==
The corresponding idea in [[module theory]] is that of a '''graded module''', namely a left [[module (mathematics)|module]] ''M'' over a graded ring ''R'' such that
: <math>M = \bigoplus_{i\in \mathbb{N}}M_i ,</math>
and
: <math>R_iM_j \subseteq M_{i+j}</math>
for every {{mvar|i}} and {{mvar|j}}.

Examples:
* A [[graded vector space]] is an example of a graded module over a [[field (mathematics)|field]] (with the field having trivial grading).
* 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 (ring theory)|annihilator]] of a graded module is a homogeneous ideal.
* Given an ideal ''I'' in a commutative ring ''R'' and an ''R''-module ''M'', the direct sum <math diaply=inline>\bigoplus_{n=0}^{\infty} I^n M/I^{n+1} M</math> is a graded module over the associated graded ring <math display=inline>\bigoplus_0^{\infty} I^n/I^{n+1}</math>.

A ''morphism'' <math>f: N \to M</math> of graded modules, called a '''graded morphism''' or ''graded homomorphism'' , is a [[Module homomorphism|homomorphism]] of the underlying modules that respects grading; i.e., {{tmath|1= 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 map|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 {{tmath|1= N_i = N \cap M_i }}. The [[kernel (algebra)|kernel]] and the [[image (mathematics)|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 (ring theory)|center]] is the same as to give the structure of a graded algebra to the latter ring.

Given a graded module <math>M</math>, the <math>\ell</math>-twist of <math>M</math> is a graded module defined by <math>M(\ell)_n = M_{n+\ell}</math> (cf. [[Serre's twisting sheaf]] in [[algebraic geometry]]).

Let ''M'' and ''N'' be graded modules. If <math>f\colon M \to N</math> is a morphism of modules, then ''f'' is said to have degree ''d'' if <math>f(M_n) \subseteq N_{n+d}</math>. An [[exterior derivative]] of [[differential form]]s in [[differential geometry]] is an example of such a morphism having degree 1.

== Invariants of graded modules ==

Given a graded module ''M'' over a commutative graded ring ''R'', one can associate the [[formal power series]] {{tmath|1= P(M, t) \in \Z[\![t]\!] }}:
: <math>P(M, t) = \sum \ell(M_n) t^n</math>
(assuming <math>\ell(M_n)</math> are finite.) It is called the [[Hilbert–Poincaré series]] of ''M''.

A graded module is said to be finitely generated if the underlying module is [[finitely generated module|finitely generated]]. The generators may be taken to be homogeneous (by replacing the generators by their homogeneous parts.)

Suppose ''R'' is a [[polynomial ring]] {{tmath|1= k[x_0, \dots, x_n] }}, ''k'' a field, and ''M'' a finitely generated graded module over it. Then the function <math>n \mapsto \dim_k M_n</math> 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 ==
{{seealso|Graded Lie algebra}}
An [[associative 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, <math>R\subseteq A_0</math> and the graded pieces <math>A_i</math> are ''R''-modules.

In the case where the ring ''R'' is also a graded ring, then one requires that 
: <math>R_iA_j \subseteq A_{i+j}</math>

In other words, we require ''A'' to be a graded left module over ''R''.

Examples of graded algebras are common in mathematics:
* [[Polynomial ring]]s. The homogeneous elements of degree ''n'' are exactly the homogeneous polynomials of degree ''n''.
* The [[tensor algebra]] <math>T^{\bullet} V</math> of a [[vector space]] ''V''. The homogeneous elements of degree ''n'' are the [[tensor]]s of order ''n'', {{tmath|1= T^{n} V }}.
* The [[exterior algebra]] <math>\textstyle\bigwedge\nolimits^{\bullet} V</math> and the [[symmetric algebra]] <math>S^{\bullet} V</math> are also graded algebras.
* The [[cohomology ring]] <math>H^{\bullet} </math> in any [[cohomology theory]] is also graded, being the direct sum of the cohomology groups {{tmath|1= H^n }}.

Graded algebras are much used in [[commutative algebra]] and [[algebraic geometry]], [[homological algebra]], and [[algebraic topology]]. One example is the close relationship between [[homogeneous polynomial]]s and [[projective varieties]] (cf. [[Homogeneous coordinate ring]].)

== ''G''-graded rings and algebras ==
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 decomposition
: <math>R = \bigoplus_{i\in G}R_i </math>
such that
: <math> R_i R_j \subseteq R_{i \cdot j}. </math>
Elements of ''R'' that lie inside <math>R_i</math> for some <math>i \in G</math> are said to be '''homogeneous''' of '''grade''' ''i''.

The previously defined notion of "graded ring" now becomes the same thing as an <math>\N</math>-graded ring, where <math>\N</math> is the monoid of [[natural number]]s under addition. The definitions for graded modules and algebras can also be extended this way replacing the indexing set <math>\N</math> with any monoid ''G''.

Remarks:
* If we do not require that the ring have an [[identity element]], [[semigroup]]s may replace monoids.

Examples:
* A [[group (mathematics)|group]] naturally grades the corresponding [[group ring]]; similarly, [[monoid ring]]s are graded by the corresponding monoid.
* An (associative) [[superalgebra]] is another term for a [[cyclic group|<math>\Z_2</math>]]-graded algebra. Examples include [[Clifford algebra]]s. Here the homogeneous elements are either of degree 0 (even) or 1 (odd).

=== Anticommutativity ===
Some graded rings (or algebras) are endowed with an [[anticommutative]] structure.  This notion requires a [[Monoid#Monoid homomorphisms|homomorphism]] of the monoid of the gradation into the additive monoid of <math>\Z/2\Z</math>, the field with two elements.  Specifically, a '''signed monoid''' consists of a pair <math>(\Gamma, \varepsilon)</math> where <math>\Gamma</math> is a monoid and <math>\varepsilon \colon \Gamma \to\Z/2\Z</math> is a homomorphism of additive monoids.  An '''anticommutative <math>\Gamma</math>-graded ring''' is a ring ''A'' graded with respect to <math>\Gamma</math> such that:
: <math>xy=(-1)^{\varepsilon (\deg x) \varepsilon (\deg y)}yx ,</math>
for all homogeneous elements ''x'' and ''y''.

=== Examples ===
* An [[exterior algebra]] is an example of an anticommutative algebra, graded with respect to the structure <math>(\Z, \varepsilon)</math> where <math>\varepsilon \colon \Z \to\Z/2\Z</math> is the quotient map.
* A [[supercommutative algebra]] (sometimes called a '''skew-commutative associative ring''') is the same thing as an anticommutative <math>(\Z, \varepsilon)</math>-graded algebra, where <math>\varepsilon</math> is the [[identity map]] of the additive structure of {{tmath|1= \Z/2\Z }}.

== Graded monoid ==
Intuitively, a graded [[monoid]] is the subset of a graded ring, <math display=inline>\bigoplus_{n\in \mathbb N_0}R_n</math>, generated by the <math>R_n</math>'s, without using the additive part. That is, the set of elements of the graded monoid is <math>\bigcup_{n\in\mathbb N_0}R_n</math>.

Formally, a graded monoid<ref>{{cite book | last=Sakarovitch | first=Jacques | title=Elements of automata theory | translator-first=Reuben|translator-last=Thomas | publisher=Cambridge University Press | year=2009 | isbn=978-0-521-84425-3 | zbl=1188.68177 | chapter = Part II: The power of algebra | page=384 }}</ref> is a monoid <math>(M,\cdot)</math>, with a gradation function <math>\phi:M\to\mathbb N_0</math> such that <math>\phi(m\cdot m')=\phi(m)+\phi(m')</math>. Note that the gradation of <math>1_M</math> is necessarily 0. Some authors request furthermore that <math>\phi(m)\ne 0</math>
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 <math>g^n</math> where ''g'' is the [[cardinality]] of a [[generator (monoid)|generating set]] ''G'' of the monoid. Therefore the number of elements of gradation ''n'' or less is at most <math>n+1</math> (for <math>g=1</math>) or <math display=inline>\frac{g^{n+1}-1}{g-1}</math> else. Indeed, each such element is the product of at most ''n'' elements of ''G'', and only <math display=inline>\frac{g^{n+1}-1}{g-1}</math> 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 [[Divisibility_(ring_theory)#Definition|divisor]] in such a graded monoid.

=== Power series indexed by a graded monoid ===
{{see also|Novikov ring}} 
These notions allow us to extend the notion of [[power series ring]]. Instead of the indexing family being <math>\mathbb N</math>, 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 <math>(K,+_K,\times_K)</math> be an arbitrary [[semiring]] and <math>(R,\cdot,\phi)</math> a graded monoid. Then <math>K\langle\langle R\rangle\rangle</math> 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 <math>s,s'\in K\langle\langle R\rangle\rangle</math> is defined pointwise, it is the function sending <math>m\in R</math> to <math>s(m)+_Ks'(m)</math>, and the product is the function sending <math>m\in R</math> to the infinite sum <math>\sum_{p,q \in R \atop p \cdot q=m}s(p)\times_K s'(q)</math>. This sum is correctly defined (i.e., finite) because, for each ''m'', there are only a finite number of pairs {{nowrap|(''p'', ''q'')}} such that {{nowrap|1=''pq'' = ''m''}}.

=== Free monoid ===
In [[formal language theory]], given an [[alphabet (formal languages)|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 ==
* [[Associated graded ring]]
* [[Differential graded algebra]]
* [[Filtered algebra]], a generalization
* [[Graded (mathematics)]]
* [[Graded category]]
* [[Graded vector space]]
* [[Tensor algebra]]<!-- if I remember correctly, any graded algebra is a quotient of a tensor algebra. -->
* [[Differential graded module]]

== Notes ==
=== Citations ===
{{reflist}}
=== References ===
{{refbegin}}
* {{Lang Algebra}}.
* {{cite book |author-link=Nicolas Bourbaki |first=N. |last=Bourbaki |chapter=Ch. 1–3, 3 §3 |title=Algebra I |publisher= Springer|location= |year=1974 |isbn=978-3-540-64243-5 }}
* {{cite journal |first=J. |last=Steenbrink |title=Intersection form for quasi-homogeneous singularities  |journal=Compositio Mathematica |volume=34 |issue=2 |pages=211–223 See p. 211 |year=1977 |issn=0010-437X |url=http://archive.numdam.org/article/CM_1977__34_2_211_0.pdf}}
* {{cite book |first=H. |last=Matsumura |title=Commutative Ring Theory |url=https://books.google.com/books?id=J68-BAAAQBAJ |date=1989 |publisher=Cambridge University Press |isbn=978-1-107-71712-1 |translator-first=M. |translator-last=Reid |edition=2nd |series=Cambridge Studies in Advanced Mathematics |volume=8 |chapter=5 Dimension theory §S3 Graded rings, the Hilbert function and the Samuel function}}
{{refend}}

[[Category:Algebras]]
[[Category:Ring theory]]