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 -algebra.
A nonzero element of is said to be homogeneous of degree. By definition of a direct sum, every nonzero element of can be uniquely written as a sum where each is either 0 or homogeneous of degree . The nonzero are the homogeneous components of .
Some basic properties are:
is a subring of ; in particular, the multiplicative identity is an homogeneous element of degree zero.
For any , is a two-sided -module, and the direct sum decomposition is a direct sum of -modules.
An ideal is homogeneous, if for every , the homogeneous components of also belong to (Equivalently, if it is a graded submodule of ; see § Graded module.) The intersection of a homogeneous ideal with is an -submodule of called the homogeneous part of degree of . A homogeneous ideal is the direct sum of its homogeneous parts.
If is a two-sided homogeneous ideal in , then is also a graded ring, decomposed as
where is the homogeneous part of degree of .
Any (non-graded) ring R can be given a gradation by letting , and for i ≠ 0. This is called the trivial gradation on R.
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 is a graded module over the associated graded ring .
A morphism between graded modules, called a graded morphism, is a morphism of underlying modules that respects grading; i.e., . 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 . 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.
The above definitions have been generalized to rings graded using any monoidG as an index set. A G-graded ringR is a ring with a direct sum decomposition
Elements of R that lie inside for some are said to be homogeneous of gradei.
The previously defined notion of "graded ring" now becomes the same thing as an -graded ring, where 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 with any monoid G.
If we do not require that the ring have an identity element, semigroups may replace monoids.
An (associative) superalgebra is another term for a -graded algebra. Examples include Clifford algebras. Here the homogeneous elements are either of degree 0 (even) or 1 (odd).
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 , the field with two elements. Specifically, a signed monoid consists of a pair where is a monoid and is a homomorphism of additive monoids. An anticommutative -graded ring is a ring A graded with respect to Γ such that:
for all homogeneous elements x and y.
An exterior algebra is an example of an anticommutative algebra, graded with respect to the structure where is the quotient map.
A supercommutative algebra (sometimes called a skew-commutative associative ring) is the same thing as an anticommutative -graded algebra, where is the identity map of the additive structure of .
Intuitively, a graded monoid is the subset of a graded ring, , generated by the 's, without using the additive part. That is, the set of elements of the graded monoid is .
Formally, a graded monoid is a monoid , with a gradation function such that . Note that the gradation of is necessarily 0. Some authors request furthermore that
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 where g is the cardinality of a generating setG of the monoid. Therefore the number of elements of gradation n or less is at most (for ) or else. Indeed, each such element is the product of at most n elements of G, and only 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 monoidEdit
This notions allows to extends the notion of power series ring. Instead of having the indexing family being , 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 be an arbitrary semiring and a graded monoid. Then 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 is defined pointwise, it is the function sending to , and the product is the function sending to the infinite sum . 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.
Steenbrink, J. (1977). "Intersection form for quasi-homogeneous singularities" (PDF). Compositio Mathematica. 34 (2): 211–223 See p. 211. ISSN 0010-437X.
Matsumura, H. (1989). "5 Dimension theory §S3 Graded rings, the Hilbert function and the Samuel function". Commutative Ring Theory. Cambridge Studies in Advanced Mathematics. Vol. 8. Translated by Reid, M. (2nd ed.). Cambridge University Press. ISBN 978-1-107-71712-1.