Let $G$ be a finite group and $R$ be a ring. There is a well-known description of the center of the group ring: $Z(R[G])$ is a free $Z(R)$-module, the basis consists of all $\sum_{g \in K} g$, where $K$ is a conjugacy class in $G$.
But $Z(R[G])$ is a commutative ring, or even better, a commutative $Z(R)$-algebra. The above description is not "compatible" with the multiplication; for example the basis is not closed under multiplication (as far as I can tell, maybe I am wrong).
Question 1. So what is a ring-theoretic representation of $Z(R[G])$? The best case would be something like $Z(R)[M]$ for some monoid $M$, but probably this is way too optimistic.
Question 2. Can we also describe the center when $G$ is not finite?