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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in particular
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, a graded ring is a ring such that the underlying additive group is a
direct sum of abelian groups The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
$R_i$ such that $R_i R_j \subseteq R_$. The index set is usually the set of nonnegative
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s or the set of integers, but can be any
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. 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 $\Z$-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.

# 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 that is decomposed into a direct sum :$R = \bigoplus_^\infty R_n = R_0 \oplus R_1 \oplus R_2 \oplus \cdots$ of additive groups, such that :$R_mR_n \subseteq R_$ for all nonnegative integers $m$ and $n$. A nonzero element of $R_n$ is said to be ''homogeneous'' of ''degree'' $n$. By definition of a direct sum, every nonzero element $a$ of $R$ can be uniquely written as a sum $a=a_0+a_1+\cdots +a_n$ where each $a_i$ is either 0 or homogeneous of degree $i$. The nonzero $a_i$ are the ''homogeneous components'' of $a$. Some basic properties are: *$R_0$ is a subring of $R$; in particular, the multiplicative identity $1$ is an homogeneous element of degree zero. *For any $n$, $R_n$ is a two-sided $R_0$-
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...
, and the direct sum decomposition is a direct sum of $R_0$-modules. * $R$ is an associative $R_0$-algebra. An
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
$I\subseteq R$ is ''homogeneous'', if for every $a \in I$, the homogeneous components of $a$ also belong to $I.$ (Equivalently, if it is a graded submodule of $R$; see .) The intersection of a homogeneous ideal $I$ with $R_n$ is an $R_0$- submodule of $R_n$ called the ''homogeneous part'' of degree $n$ of $I$. A homogeneous ideal is the direct sum of its homogeneous parts. If $I$ is a two-sided homogeneous ideal in $R$, then $R/I$ is also a graded ring, decomposed as : $R/I = \bigoplus_^\infty R_n/I_n,$ where $I_n$ is the homogeneous part of degree $n$ of $I$.

# Basic examples

*Any (non-graded) ring ''R'' can be given a gradation by letting $R_0=R$, and $R_i=0$ for ''i'' ≠ 0. This is called the trivial gradation on ''R''. *The
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
s 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 $\Z$-graded ring. *If ''I'' is an ideal in a commutative ring ''R'', then $\bigoplus_^ I^n/I^$ 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 In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, ''H'' ''i''(''X''; ''R'') the ''i''th cohomology group with coefficients in a ring ''R''. Then ''H'' *(''X''; ''R''), the cohomology ring of ''X'' with coefficients in ''R'', is a graded ring whose underlying group is $\bigoplus_^\infty H^i\left(X; R\right)$ with the multiplicative structure given by the
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutati ...
.

The corresponding idea in module theory is that of a graded module, namely a left
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...
''M'' over a graded ring ''R'' such that also :$M = \bigoplus_M_i ,$ and :$R_iM_j \subseteq M_.$ 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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
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 $\bigoplus_^ I^n M/I^ M$ is a graded module over the associated graded ring $\bigoplus_0^ I^n/I^$. A morphism $f: N \to M$ between graded modules, called a graded morphism, is a morphism of underlying modules that respects grading; i.e., $f\left(N_i\right) \subseteq M_i$. 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 $N_i = N \cap M_i$. The kernel and the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
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 $M$, the $\ell$-twist of $M$ is a graded module defined by $M\left(\ell\right)_n = M_$. (cf. Serre's twisting sheaf in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
.) Let ''M'' and ''N'' be graded modules. If $f\colon M \to N$ is a morphism of modules, then ''f'' is said to have degree ''d'' if $f\left(M_n\right) \subseteq N_$. An
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
of
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
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 In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...

An
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
''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, $R\subseteq A_0$ and the graded pieces $A_i$ are ''R''-modules. In the case where the ring ''R'' is also a graded ring, then one requires that :$R_iA_j \subseteq A_$ In other words, we require ''A'' to be a graded left module over ''R''. Examples of graded algebras are common in mathematics: *
Polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
s. The homogeneous elements of degree ''n'' are exactly the homogeneous polynomials of degree ''n''. * The tensor algebra $T^ V$ of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''V''. The homogeneous elements of degree ''n'' are the
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
s of order ''n'', $T^ V$. * The exterior algebra $\textstyle\bigwedge\nolimits^ V$ and the symmetric algebra $S^ V$ are also graded algebras. * The cohomology ring $H^$ in any cohomology theory is also graded, being the direct sum of the cohomology groups $H^n$. Graded algebras are much used in
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
and
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, homological algebra, and
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
. One example is the close relationship between
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
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 In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoid ...
''G'' as an index set. A ''G''-graded ring ''R'' is a ring with a direct sum decomposition :$R = \bigoplus_R_i$ such that :$R_i R_j \subseteq R_.$ Elements of ''R'' that lie inside $R_i$ for some $i \in G$ are said to be homogeneous of grade ''i''. The previously defined notion of "graded ring" now becomes the same thing as an $\N$-graded ring, where $\N$ is the monoid of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s under addition. The definitions for graded modules and algebras can also be extended this way replacing the indexing set $\N$ with any monoid ''G''. Remarks: *If we do not require that the ring have an identity element, semigroups may replace monoids. Examples: *A group naturally grades the corresponding group ring; similarly, monoid rings are graded by the corresponding monoid. *An (associative) superalgebra is another term for a $\Z_2$-graded algebra. Examples include Clifford algebras. Here the homogeneous elements are either of degree 0 (even) or 1 (odd).

## Anticommutativity

Some graded rings (or algebras) are endowed with an
anticommutative In mathematics, anticommutativity is a specific property of some non- commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswappe ...
structure. This notion requires a
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
of the monoid of the gradation into the additive monoid of $\Z/2\Z$, the field with two elements. Specifically, a signed monoid consists of a pair $\left(\Gamma, \varepsilon\right)$ where $\Gamma$ is a monoid and $\varepsilon \colon \Gamma \to\Z/2\Z$ is a homomorphism of additive monoids. An anticommutative $\Gamma$-graded ring is a ring ''A'' graded with respect to Γ such that: :$xy=\left(-1\right)^yx ,$ for all homogeneous elements ''x'' and ''y''.

## Examples

*An exterior algebra is an example of an anticommutative algebra, graded with respect to the structure $\left(\Z, \varepsilon\right)$ where $\varepsilon \colon \Z \to\Z/2\Z$ is the quotient map. *A supercommutative algebra (sometimes called a skew-commutative associative ring) is the same thing as an anticommutative $\left(\Z, \varepsilon\right)$-graded algebra, where $\varepsilon$ is the
identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
of the additive structure of $\Z/2\Z$.

monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoid ...
is the subset of a graded ring, $\bigoplus_R_n$, generated by the $R_n$'s, without using the additive part. That is, the set of elements of the graded monoid is $\bigcup_R_n$. Formally, a graded monoid is a monoid $\left(M,\cdot\right)$, with a gradation function $\phi:M\to\mathbb N_0$ such that $\phi\left(m\cdot m\text{'}\right)=\phi\left(m\right)+\phi\left(m\text{'}\right)$. Note that the gradation of $1_M$ is necessarily 0. Some authors request furthermore that $\phi\left(m\right)\ne 0$ 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 $g^n$ 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 $n+1$ (for $g=1$) or $\frac$ else. Indeed, each such element is the product of at most ''n'' elements of ''G'', and only $\frac$ 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 mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
in such a graded monoid.

## Power series indexed by a graded monoid

This notions allows to extends the notion of power series ring. Instead of having the indexing family being $\mathbb N$, 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 $\left(K,+_K,\times_K\right)$ be an arbitrary semiring and $\left(R,\cdot,\phi\right)$ a graded monoid. Then $K\langle\langle R\rangle\rangle$ 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 $s,s\text{'}\in K\langle\langle R\rangle\rangle$ is defined pointwise, it is the function sending $m\in R$ to $s\left(m\right)+_Ks\text{'}\left(m\right)$, and the product is the function sending $m\in R$ to the infinite sum $\sum_s\left(p\right)\times_K s\text{'}\left(q\right)$. This sum is correctly defined (i.e., finite) because, for each ''m'', there are only a finite number of pairs such that .

## Example

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.