Is the module of homomorphisms between graded modules also a graded module?

Question

If $M,N$ are graded modules over a graded ring $R$, then is $\operatorname{Hom}_{R}(M,N)$ also a graded module and how?

Answer 1

If by $\text{Hom}_R$ you mean graded homomorphisms (those that preserve the grading), then no. However, there is a "graded Hom" where the $i^{th}$ graded component consists of homomorphisms which raise degree by $i$, and the zeroth graded component of the graded Hom is the ordinary Hom. The general keyword here is internal Hom.