For any commutative unital ring $R$ , $R[X]$ has a standard $\mathbb N \cup \{0\}$ grading where the $n$-th component is $R_n:= X^nR[X]$ ; however , given a commutative , unital , graded ring $R$ , I am not sure how to define a canonical grading for $R[X]$ which respects the grading of $R$ in some sense in that the usual inclusion map $i: R \to R[X]$ should be a graded ring homomorphism . So my question is :
Let $R$ be a commutative unital $G$- graded ring , where $G$ is a monoid . Does there exist a gradation on $R[X]$ such that the usual inclusion map $i: R \to R[X]$ is a graded ring homomorphism ? Does there also exist a gradation on $R[X]$ satisfying the previous condition and also satisfying that $I[X]$ is a graded ideal of $R[X]$ whenever $I$ is a graded ideal of $R$ ? If not true in general , then is there any condition on $G$ which answers at least one of the questions in affirmative ?