In

^{ ''i''}(''X''; ''R'') the ''i''th ^{ *}(''X''; ''R''), the

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 te ...

, a graded ring is a ring such that the underlying additive group
An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation.
This terminology is widely used with structur ...

is a direct sum of abelian groups $R\_i$ such that $R\_i\; R\_j\; \backslash subseteq\; R\_$. 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.
Monoids a ...

. 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
In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces.
Integer gradation
Let \mathbb be t ...

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 $\backslash Z$-algebra.
The associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacemen ...

is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to non-associative algebra
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure ''A'' is a non-associative algebra over a field ''K'' if ...

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 that is decomposed into adirect sum
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 ...

:$R\; =\; \backslash bigoplus\_^\backslash infty\; R\_n\; =\; R\_0\; \backslash oplus\; R\_1\; \backslash oplus\; R\_2\; \backslash oplus\; \backslash cdots$
of
additive group
An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation.
This terminology is widely used with structur ...

s, such that
:$R\_mR\_n\; \backslash subseteq\; R\_$
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 $a=a\_0+a\_1+\backslash cdots\; +a\_n$ 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
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...

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, 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 considere ...

$I\backslash subseteq\; R$ is ''homogeneous'', if for every $a\; \backslash in\; I$, 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
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the m ...

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
: $R/I\; =\; \backslash bigoplus\_^\backslash infty\; R\_n/I\_n,$
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''. *Thepolynomial 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 variab ...

$R\; =\; k;\; href="/html/ALL/l/\_1,\_\backslash ldots,\_t\_n.html"\; ;"title="\_1,\; \backslash ldots,\; t\_n">\_1,\; \backslash ldots,\; t\_n$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
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...

''R''. Then the localization of ''R'' with respect to ''S'' is a $\backslash Z$-graded ring.
*If ''I'' is an ideal in a commutative ring
In mathematics, a commutative ring is a ring in which 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 ...

''R'', then $\backslash bigoplus\_^\; I^n/I^$ is a graded ring called the associated graded ring In mathematics, the associated graded ring of a ring ''R'' with respect to a proper ideal ''I'' is the graded ring:
:\operatorname_I R = \oplus_^\infty I^n/I^.
Similarly, if ''M'' is a left ''R''-module, then the associated graded module is the gr ...

of ''R'' along ''I''; geometrically, it is the coordinate ring
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea ...

of the normal cone
In algebraic geometry, the normal cone C_XY of a subscheme X of a scheme Y is a scheme analogous to the normal bundle or tubular neighborhood in differential geometry.
Definition
The normal cone or C_ of an embedding , defined by some sheaf of i ...

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 poi ...

, ''H''cohomology group
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

with coefficients in a ring ''R''. Then ''H''cohomology ring In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually un ...

of ''X'' with coefficients in ''R'', is a graded ring whose underlying group is $\backslash bigoplus\_^\backslash infty\; H^i(X;\; R)$ 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 commuta ...

.
Graded module

The corresponding idea in module theory is that of a graded module, namely a left module ''M'' over a graded ring ''R'' such that also :$M\; =\; \backslash bigoplus\_M\_i\; ,$ and :$R\_iM\_j\; \backslash subseteq\; M\_.$ Example: agraded vector space
In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces.
Integer gradation
Let \mathbb be t ...

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 $\backslash bigoplus\_^\; I^n\; M/I^\; M$ is a graded module over the associated graded ring $\backslash bigoplus\_0^\; I^n/I^$.
A morphism $f:\; N\; \backslash to\; M$ between graded modules, called a graded morphism, is a morphism of underlying modules that respects grading; i.e., $f(N\_i)\; \backslash subseteq\; M\_i$. 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 $N\_i\; =\; N\; \backslash cap\; M\_i$. The kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learni ...

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-dimension ...

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
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentricity ...

is the same as to give the structure of a graded algebra to the latter ring.
Given a graded module $M$, the $\backslash ell$-twist of $M$ is a graded module defined by $M(\backslash ell)\_n\; =\; M\_$. (cf. Serre's twisting sheaf
In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not functo ...

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 $f\backslash colon\; M\; \backslash to\; N$ is a morphism of modules, then ''f'' is said to have degree ''d'' if $f(M\_n)\; \backslash subseteq\; N\_$. 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 r ...

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 applications, ...

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 mul ...

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 theformal 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 sum ...

$P(M,\; t)\; \backslash in\; \backslash Z;\; href="/html/ALL/l/!;\; \_;"title="![t">![t$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 variab ...

$k;\; href="/html/ALL/l/\_0,\_\backslash dots,\_x\_n.html"\; ;"title="\_0,\; \backslash dots,\; x\_n">\_0,\; \backslash dots,\; x\_n$integer-valued polynomial In mathematics, an integer-valued polynomial (also known as a numerical polynomial) P(t) is a polynomial whose value P(n) is an integer for every integer ''n''. Every polynomial with integer coefficients is integer-valued, but the converse is not tr ...

for large ''n'' called the Hilbert polynomial of ''M''.
Graded algebra

Analgebra
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, $R\backslash subseteq\; A\_0$ 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
:$R\_iA\_j\; \backslash subseteq\; A\_$
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 variab ...

s. The homogeneous elements of degree ''n'' are exactly the homogeneous polynomials of degree ''n''.
* The tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of be ...

$T^\; V$ of a vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...

''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 tenso ...

s of order ''n'', $T^\; V$.
* The exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...

$\backslash textstyle\backslash bigwedge\backslash nolimits^\; V$ and the symmetric algebra
In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...

$S^\; V$ are also graded algebras.
* The cohomology ring In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually un ...

$H^$ in any cohomology theory
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

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. Promine ...

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
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topol ...

, 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 u ...

. 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
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...

(cf. Homogeneous coordinate ring In algebraic geometry, the homogeneous coordinate ring ''R'' of an algebraic variety ''V'' given as a subvariety of projective space of a given dimension ''N'' is by definition the quotient ring
:''R'' = ''K'' 'X''0, ''X''1, ''X''2, ..., ''X'N' ...

.)
''G''-graded rings and algebras

The above definitions have been generalized to rings graded using anymonoid
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.
Monoids a ...

''G'' as an index set. A ''G''-graded ring ''R'' is a ring with a direct sum decomposition
:$R\; =\; \backslash bigoplus\_R\_i$
such that
:$R\_i\; R\_j\; \backslash subseteq\; R\_.$
Elements of ''R'' that lie inside $R\_i$ for some $i\; \backslash in\; G$ are said to be homogeneous of grade ''i''.
The previously defined notion of "graded ring" now becomes the same thing as an $\backslash N$-graded ring, where $\backslash N$ 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 $\backslash N$ with any monoid ''G''.
Remarks:
*If we do not require that the ring have an identity element, semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...

s may replace monoids.
Examples:
*A group naturally grades the corresponding group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the give ...

; similarly, monoid ring
In abstract algebra, a monoid ring is a ring constructed from a ring and a monoid, just as a group ring is constructed from a ring and a group.
Definition
Let ''R'' be a ring and let ''G'' be a monoid. The monoid ring or monoid algebra of ''G'' ...

s are graded by the corresponding monoid.
*An (associative) superalgebra
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.
T ...

is another term for a $\backslash Z\_2$-graded algebra. Examples include Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hype ...

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 ahomomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...

of the monoid of the gradation into the additive monoid of $\backslash Z/2\backslash Z$, the field with two elements. Specifically, a signed monoid consists of a pair $(\backslash Gamma,\; \backslash varepsilon)$ where $\backslash Gamma$ is a monoid and $\backslash varepsilon\; \backslash colon\; \backslash Gamma\; \backslash to\backslash Z/2\backslash Z$ is a homomorphism of additive monoids. An anticommutative $\backslash Gamma$-graded ring is a ring ''A'' graded with respect to Γ such that:
:$xy=(-1)^yx\; ,$
for all homogeneous elements ''x'' and ''y''.
Examples

*Anexterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...

is an example of an anticommutative algebra, graded with respect to the structure $(\backslash Z,\; \backslash varepsilon)$ where $\backslash varepsilon\; \backslash colon\; \backslash Z\; \backslash to\backslash Z/2\backslash Z$ is the quotient map.
*A supercommutative algebra
In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements ''x'', ''y'' we have
:yx = (-1)^xy ,
where , ''x'', denotes the grade of the element and is 0 or 1 ...

(sometimes called a skew-commutative associative ring) is the same thing as an anticommutative $(\backslash Z,\; \backslash varepsilon)$-graded algebra, where $\backslash varepsilon$ is the identity map of the additive structure of $\backslash Z/2\backslash Z$.
Graded monoid

Intuitively, a gradedmonoid
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.
Monoids a ...

is the subset of a graded ring, $\backslash bigoplus\_R\_n$, generated by the $R\_n$'s, without using the additive part. That is, the set of elements of the graded monoid is $\backslash bigcup\_R\_n$.
Formally, a graded monoid is a monoid $(M,\backslash cdot)$, with a gradation function $\backslash phi:M\backslash to\backslash mathbb\; N\_0$ such that $\backslash phi(m\backslash cdot\; m\text{'})=\backslash phi(m)+\backslash phi(m\text{'})$. Note that the gradation of $1\_M$ is necessarily 0. Some authors request furthermore that $\backslash phi(m)\backslash ne\; 0$
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
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied t ...

''G'' of the monoid. Therefore the number of elements of gradation ''n'' or less is at most $n+1$ (for $g=1$) or $\backslash frac$ else. Indeed, each such element is the product of at most ''n'' elements of ''G'', and only $\backslash frac$ 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 $\backslash mathbb\; N$, 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 $(K,+\_K,\backslash times\_K)$ be an arbitrary semiring and $(R,\backslash cdot,\backslash phi)$ a graded monoid. Then $K\backslash langle\backslash langle\; R\backslash rangle\backslash rangle$ 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 $s,s\text{'}\backslash in\; K\backslash langle\backslash langle\; R\backslash rangle\backslash rangle$ is defined pointwise, it is the function sending $m\backslash in\; R$ to $s(m)+\_Ks\text{'}(m)$, and the product is the function sending $m\backslash in\; R$ to the infinite sum $\backslash sum\_s(p)\backslash times\_K\; s\text{'}(q)$. This sum is correctly defined (i.e., finite) because, for each ''m'', there are only a finite number of pairs such that .Example

Informal language theory
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.
The alphabet of a formal language consists of sym ...

, given an alphabet ''A'', the free monoid In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero ele ...

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 In mathematics, the associated graded ring of a ring ''R'' with respect to a proper ideal ''I'' is the graded ring:
:\operatorname_I R = \oplus_^\infty I^n/I^.
Similarly, if ''M'' is a left ''R''-module, then the associated graded module is the gr ...

* Differential graded algebra
* Filtered algebra, a generalization
* Graded (mathematics)
In mathematics, the term “graded” has a number of meanings, mostly related:
In abstract algebra, it refers to a family of concepts:
* An algebraic structure X is said to be I-graded for an index set I if it has a gradation or grading, i.e. a d ...

* Graded category
* Graded vector space
In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces.
Integer gradation
Let \mathbb be t ...

* Tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of be ...

* Differential graded module
References

* . * * {{refend Algebras Ring theory