Graded ring

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 formula_2. 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 formula_3-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 supposed 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
of 
additive groups, such that 
for all nonnegative integers formula_6 and formula_7.
A nonzero element of formula_8 is said to be "homogeneous" of "degree" formula_7. By definition of a direct sum, every nonzero element formula_10 of formula_11 can be uniquely written as a sum formula_12 where each formula_13 is either 0 or homogeneous of degree formula_14. The nonzero formula_13 are the "homogeneous components" of formula_10.
Some basic properties are:
An ideal formula_26 is "homogeneous", if for every formula_27, the homogeneous components of formula_10 also belong to formula_29 (Equivalently, if it is a graded submodule of formula_11; see .) The intersection of a homogeneous ideal formula_31 with formula_8 is an formula_17-submodule of formula_8 called the "homogeneous part" of degree formula_7 of formula_31. A homogeneous ideal is the direct sum of its homogeneous parts.
If formula_31 is a two-sided homogeneous ideal in formula_11, then formula_39 is also a graded ring, decomposed as
where formula_41 is the homogeneous part of degree formula_7 of formula_31.
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
and
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 formula_53 is a graded module over the associated graded ring formula_49.
A morphism formula_55 between graded modules, called a graded morphism, is a morphism of underlying modules that respects grading; i.e., formula_56. 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 formula_57. 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_58, the formula_59-twist of formula_58 is a graded module defined by formula_61. (cf. Serre's twisting sheaf in algebraic geometry.)
Let "M" and "N" be graded modules. If formula_62 is a morphism of modules, then "f" is said to have degree "d" if formula_63. 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 formula_64:
(assuming formula_66 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 formula_67, "k" a field, and "M" a finitely generated graded module over it. Then the function formula_68 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 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_69 and the graded pieces formula_70 are "R"-modules.
In the case where the ring "R" is also a graded ring, then one requires that 
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
such that
Elements of "R" that lie inside formula_1 for some formula_81 are said to be homogeneous of grade "i".
The previously defined notion of "graded ring" now becomes the same thing as an formula_82-graded ring, where formula_82 is the monoid of non-negative integers under addition. The definitions for graded modules and algebras can also be extended this way replacing the indexing set formula_82 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_86, the field with two elements. Specifically, a signed monoid consists of a pair formula_87 where formula_88 is a monoid and formula_89 is a homomorphism of additive monoids. An anticommutative formula_88-graded ring is a ring "A" graded with respect to Γ such that:
for all homogeneous elements "x" and "y".
Graded monoid.
Intuitively, a graded monoid is the subset of a graded ring, formula_97, generated by the formula_8's, without using the additive part. That is, the set of elements of the graded monoid is formula_99.
Formally, a graded monoid is a monoid formula_100, with a gradation function formula_101 such that formula_102. Note that the gradation of formula_103 is necessarily 0. Some authors request furthermore that formula_104
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_105 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_106 (for formula_107) or formula_108 else. Indeed, each such element is the product of at most "n" elements of "G", and only formula_108 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.
This notions allows to extends the notion of power series ring. Instead of having the indexing family being formula_110, 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_111 be an arbitrary semiring and formula_112 a graded monoid. Then formula_113 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_114 is defined point-wise, it is the function sending formula_115 to formula_116. And the product is the function sending formula_115 to the infinite sum formula_118. This sum is correctly defined (i.e., finite) because, for each "m", only a finite number of pairs such that exist.
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.