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 such that . 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 -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
of
additive groups, such that
for all nonnegative integers and .
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. is an associative -algebra.
An ideal is homogeneous, if for every , the homogeneous components of also belong to (Equivalently, if it is a graded submodule of ; see .) 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 .
Basic examples
Any (non-graded) ring R can be given a gradation by letting , and for i ≠ 0. This is called the trivial gradation on R. The polynomial ring is graded by degree: it is a direct sum of consisting of homogeneous polynomials 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 -graded ring. If I is an ideal in a commutative ring R, then 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, H i(X; R) the ith 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 with the multiplicative structure given by the cup product.
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 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.
Given a graded module , the -twist of is a graded module defined by . (cf. Serre's twisting sheaf in algebraic geometry.)
Let M and N be graded modules. If is a morphism of modules, then f is said to have degree d if . An exterior derivative of differential forms in differential geometry is an example of such a morphism having degree 1.