Ring generated by character values
In characteristic zero or a prime characteristic
Suppose is a finite group and fix a characteristic that is either zero or a prime not dividing the order of . The ring generated by character values is the smallest ring containing all the character values of 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 , because characters are cyclotomic integers.
Note that this differes from the character ring, which is the ring of characters as functions.