Graded Algebra - 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

Graded modules may be considered over non-graded rings by giving the trivial gradation to the ring. This allows to consider a sequence of modules as a single graded module. This is used in homological algebra to extend to chain complexes some module constructions like direct sum or tensor product.