In mathematics, in particular abstract algebra, a graded ring is a ring that is a direct sum of abelian groups
Contents
- First properties
- Basic examples
- Graded module
- Invariants of graded modules
- Graded algebra
- G graded rings and algebras
- Anticommutativity
- Examples
- References
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 Z-algebra.
The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to a non-associative algebra as well; e.g., one can consider a graded Lie algebra.
First properties
Let
be a graded ring. Elements of any factor
Some basic properties are:
An ideal
If I is a homogeneous ideal in A, then
Basic examples
Graded module
The corresponding idea in module theory is that of a graded module, namely a left module M over a graded ring A 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,
A morphism
Remark: To give a graded morphism from a graded ring to a 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 M, the ℓ-twist of
Let M and N be graded modules. If
Invariants of graded modules
Given a graded module M over a commutative graded ring A, one can associate the formal power series
(assuming
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 A is a polynomial ring
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 grade 0). Thus, R ⊆ A0 and the Ai are R-modules.
In the case where the ring R is also a graded ring, then one requires that
and
In other words, we require A to be a left and right graded 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 gradings ring using any monoid G as an index set. A G-graded ring A is a ring with a direct sum decomposition
such that
The notion of "graded ring" now becomes the same thing as a N-graded ring, where N 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 N 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 Z/2Z, the field with two elements. Specifically, a signed monoid consists of a pair (Γ, ε) where Γ is a monoid and ε : Γ → Z/2Z is a homomorphism of additive monoids. An anticommutative Γ-graded ring is a ring A graded with respect to Γ such that:
for all homogeneous elements x and y.