Type of algebraic structure
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups formula_1 such that &amp;NoBreak;}&amp;NoBreak;. 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 &amp;NoBreak;&amp;NoBreak;-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
 formula_2 
of 
additive groups, such that 
 formula_3
for all nonnegative integers formula_4 and &amp;NoBreak;&amp;NoBreak;.
A nonzero element of formula_5 is said to be "homogeneous" of "degree" &amp;NoBreak;&amp;NoBreak;. By definition of a direct sum, every nonzero element formula_6 of formula_7 can be uniquely written as a sum formula_8 where each formula_9 is either 0 or homogeneous of degree &amp;NoBreak;&amp;NoBreak;. The nonzero formula_9 are the "homogeneous components" of &amp;NoBreak;&amp;NoBreak;.
Some basic properties are:
An ideal formula_16 is "homogeneous", if for every &amp;NoBreak;&amp;NoBreak;, the homogeneous components of formula_6 also belong to &amp;NoBreak;&amp;NoBreak;. (Equivalently, if it is a graded submodule of &amp;NoBreak;&amp;NoBreak;; see .) The intersection of a homogeneous ideal formula_18 with formula_5 is an &amp;NoBreak;&amp;NoBreak;-submodule of formula_5 called the "homogeneous part" of degree formula_13 of &amp;NoBreak;&amp;NoBreak;. A homogeneous ideal is the direct sum of its homogeneous parts.
If formula_18 is a two-sided homogeneous ideal in &amp;NoBreak;&amp;NoBreak;, then formula_23 is also a graded ring, decomposed as
 formula_24
where formula_25 is the homogeneous part of degree formula_13 of &amp;NoBreak;&amp;NoBreak;.
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
 formula_33
and
 formula_34
for every i and j.
Examples:
A "morphism" formula_37 of graded modules, called a graded morphism or "graded homomorphism" , is a homomorphism of the underlying modules that respects grading; i.e., &amp;NoBreak;&amp;NoBreak;. 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 &amp;NoBreak;&amp;NoBreak;. 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 formula_38, the formula_39-twist of formula_38 is a graded module defined by formula_41 (cf. Serre's twisting sheaf in algebraic geometry).
Let "M" and "N" be graded modules. If formula_42 is a morphism of modules, then "f" is said to have degree "d" if formula_43. An exterior derivative of differential forms 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 &amp;NoBreak;&amp;NoBreak;:
 formula_44
(assuming formula_45 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. The generators may be taken to be homogeneous (by replacing the generators by their homogeneous parts.)
Suppose "R" is a polynomial ring &amp;NoBreak;&amp;NoBreak;, "k" a field, and "M" a finitely generated graded module over it. Then the function formula_46 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.
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, formula_47 and the graded pieces formula_48 are "R"-modules.
In the case where the ring "R" is also a graded ring, then one requires that 
 formula_49
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.
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
 formula_54
such that
 formula_55
Elements of "R" that lie inside formula_1 for some formula_57 are said to be homogeneous of grade "i".
The previously defined notion of "graded ring" now becomes the same thing as an formula_58-graded ring, where formula_58 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 formula_58 with any monoid "G".
Remarks:
Examples:
Anticommutativity.
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 formula_62, the field with two elements. Specifically, a signed monoid consists of a pair formula_63 where formula_64 is a monoid and formula_65 is a homomorphism of additive monoids. An anticommutative formula_64-graded ring is a ring "A" graded with respect to formula_64 such that:
 formula_68
for all homogeneous elements "x" and "y".
Graded monoid.
Intuitively, a graded monoid is the subset of a graded ring, formula_73, generated by the formula_5's, without using the additive part. That is, the set of elements of the graded monoid is formula_75.
Formally, a graded monoid is a monoid formula_76, with a gradation function formula_77 such that formula_78. Note that the gradation of formula_79 is necessarily 0. Some authors request furthermore that formula_80
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 formula_81 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 formula_82 (for formula_83) or formula_84 else. Indeed, each such element is the product of at most "n" elements of "G", and only formula_84 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.
These notions allow us to extend the notion of power series ring. Instead of the indexing family being formula_86, 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 formula_87 be an arbitrary semiring and formula_88 a graded monoid. Then formula_89 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 formula_90 is defined pointwise, it is the function sending formula_91 to formula_92, and the product is the function sending formula_91 to the infinite sum formula_94. 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".
Free monoid.
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.
Notes.
Citations.
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References.
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