Graded Algebra

If is a graded module and there exists a degree-preserving linear map , then is called a graded algebra.

Cohomology is a graded algebra. In addition, the grading set is monoid having a compatibility relation such that if is in the grading of the algebra , and is in the grading of the algebra , then is in the grading of the algebra (where and are multiplied in , and and are multiplied in the index monoid). For example, cohomology of a space is a graded algebra over the integers (i.e., a graded ring), since if is an -dimensional cohomology class and is an -dimensional cohomology class, then the cup product is an dimensional cohomology class.

The group ring of a group over a ring is a graded -algebra with grading .