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Concept
Graded ring
Summary
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.
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 .
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