graded module

It is a well-known fact that a polynomialMathworldPlanetmathPlanetmath (over, say, ) can be written as a sum of monomials in a unique way. A monomial is a special kind of a polynomial. Unlike polynomials, the monomials can be partitioned so that the sum of any two monomial within a partition, and the productMathworldPlanetmathPlanetmathPlanetmath of any two monomials, are again monomials. As one may have guessed, one would partition the monomials by their degree. The above notion can be generalized, and the general notion is that of a graded ringMathworldPlanetmath (and a graded moduleMathworldPlanetmath).


Let R=R0R1 be a graded ring. A module M over R is said to be a graded module if


where Mi are abelianMathworldPlanetmath subgroupsMathworldPlanetmathPlanetmath of M, such that RiMjMi+j for all i,j. An element of M is said to be homogeneous of degree i if it is in Mi. The set of Mi is called a grading of M.

Whenever we speak of a graded module, the module is always assumed to be over a graded ring. As any ring R is trivially a graded ring (where Ri=R if i=0 and Ri=0 otherwise), every module M is trivially a graded module with Mi=M if i=0 and Mi=0 otherwise. However, it is customary to regard a graded module (or a graded ring) non-trivially.

If R is a graded ring, then clearly it is a graded module over itself, by setting Mi=Ri (M=R in this case). Furthermore, if M is graded over R, then so is Mz for any indeterminate z.

Example. To see a concrete example of a graded module, let us first construct a graded ring. For convenience, take any ring R, the polynomial ring S=R[x] is a graded ring, as


with Si:=Rxi. Then SiSj=(Rxm)(Rxn)Rxm+n=Si+j.

Therefore, S is a graded module over S. Similarly, the submodulesMathworldPlanetmath Sxi of S are also graded over S.

It is possible for a module over a graded ring to be graded in more than one way. Let S be defined as in the example above. Then S[y] is graded over S. One way to grade S[y] is the following:

S[y]=k=0Ak, where Ak=R[y]xk,

since SpAq=(Rxp)(R[y]xq)R[y]xp+q=Ap+q. Another way to grade S[y] is:

S[y]=k=0Bk, where Bk=i+j=kRxiyj,



Graded homomorphisms and graded submodules

Let M,N be graded modules over a (graded) ring R. A module homomorphismMathworldPlanetmath f:MN is said to be graded if f(Mi)Ni. f is a graded isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath if it is a graded module homomorphism and an isomorphism. If f is a graded isomorphism MN, then

  1. 1.

    f(Mi)=Ni. Suppose aNi and f(b)=a. Write b=bj where bjMi, and bj=0 for all but finitely many j. Then each f(bj)Nj. Since NjNi=0 if ij, f(bj)=0 if ji. Therefore b=biNi.

  2. 2.

    f-1 is graded. If af-1(Ni), then f(a)Ni=f(Mi) by the previous fact. Then f(a)=f(c) for some cMi, so a=cMi since f is one-to-one.

Suppose a graded module M has two gradings: M=Ai=Bi. The two gradings on M are said to be isomorphic if there is a graded isomorphism f on M with f(Aj)=Bj. In the example above, we see that the two gradings of S[j] are non-isomorphic.

Let N be a submodule of a graded module M (over R). We can turn N into a graded module by defining Ni=NMi. Of course, N may already be a graded module in the first place. But the two gradings on N may not be isomorphic. A submodule N of a graded module M (over R) is said to be a graded submodule of M if its grading is defined by Ni=NMi. If N is a graded submodule of M, then the injection NM is a graded homomorphism.


The above definition can be generalized, and the generalizationPlanetmathPlanetmath comes from the subscripts. The set of subscripts in the definition above is just the set of all non-negative integers (sometimes denoted ) with a binary operationMathworldPlanetmath +. It is reasonable to extend the set of subscripts from to an arbitrary set S with a binary operation *. Normally, we require that * is associative so that S is a semigroupPlanetmathPlanetmath. An R-module M is said to be S-graded if

M=sSMs such that RsMtMs*t.

Examples of S-graded modules are mainly found in modules over a semigroup ring R[S].

Remark. Graded modules, and in general the concept of grading in algebraMathworldPlanetmathPlanetmath, are an essential tool in the study of homological algebraicPlanetmathPlanetmath aspect of rings.

Title graded module
Canonical name GradedModule
Date of creation 2013-03-22 11:45:05
Last modified on 2013-03-22 11:45:05
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 18
Author CWoo (3771)
Entry type Definition
Classification msc 16W50
Classification msc 74R99
Synonym graded homomorphism
Synonym homogeneous submodule
Related topic GradedAlgebra
Defines graded module homomorphism
Defines homogeneous of degree
Defines graded submodule
Defines grading