Character ring

Definition with symbols
Given a group $$G$$, a field $$k$$, and a subring $$R$$ of $$k$$, the character ring of $$G$$ with respect to $$k$$ and $$R$$, is the ring of $$R$$-linear combinations of characters of representations of $$G$$ over $$k$$.

If $$R$$ is not specified, we assume it to be the smallest subring (viz the subring generated by 1, which is either $$\Z$$ or a prime field of the same characteristic as the whole field).