Center of a group ring?

Question

Let $R$ be a ring with unity and let $G$ be a group. Then what is the center of the group ring $R(G)$?

I feel that the center $Z(R(G))$ is $R(Z(G))$. I assumed if $x=\sum_{i=1}^{n} r_i g_i$, then it should commute with every element of $R(G)$. In particular it should commute with $g=1 \cdot g \in R(G)$

Thus we get

$$\big(\sum_{i=1}^{n} r_i g_i\big)g=g\big(\sum_{i=1}^{n} r_i g_i\big)$$

Thus I get $$g_i \cdot g=g_i \cdot g \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \forall g \in G$$

Hence $x \in Z(R(G)) $.

Is this correct?

Answer 1

The inclusion $R(Z(G))\subset Z(R(G))$ is obvious, it's the other inclusion you need to worry about.

Let $G=\mathbb S_n$. Then $R(Z(G))=R(1)$. But $x=\sum_{g\in G}g\in Z(R(G))$.

Answer 2

The centre of a group ring is spanned by sums of elements of conjugacy classes. Suppose that $C \subseteq G$ is a conjugacy class, and set $s_C = \sum_{x \in C} x$, then we have $g s_C g^{-1} = s_C$, so certainly $s_C$ is central in the group algebra. Conversely, suppose that $\sum_{x \in G} r_x x$ is central in the group algebra, then we have that $$ \sum_{x \in G} r_x x = \sum_{x \in G} r_x g x g^{-1} = \sum_{x \in G} r_{g^{-1} x g} x$$ for all $g \in G$, in other words we have $r_x = r_{g^{-1} x g}$ for all $g \in G$, so $r_x$ is constant on conjugacy classes and hence is a linear combination of the $s_C$.