Ring generated by character values

Definition

In characteristic zero or a prime characteristic

Suppose $G$ is a finite group and fix a characteristic that is either zero or a prime not dividing the order of $G$. The ring generated by character values is the smallest ring containing all the character values of $G$ over a splitting field.

Note that in prime characteristic, it is the same as the field generated by character values.

In characteristic zero, it is contained in a cyclotomic extension of the ring of integers $\mathbb {Z}$, because characters are cyclotomic integers.

Note that this differes from the character ring, which is the ring of characters as functions.