I am studying a proof of Brauer's theorem. The proof makes use of the following claim, which I haven't been able to convince myself of:
Let $G$ be a finite group and let $R[G]$ be the representation ring of $G$: i.e. the free abelian group on the irreducible representations of $G$, with multiplication given by tensor product of the irreducible representations. Let $I$ be the subset of $R[G]$ consisting of the $\mathbb{Z}$-span of representations induced from characters on elementary subgroups. Then $I$ is an ideal of $R[G]$.
Why is $I$ an ideal?
In particular, if $a\in R[G]$ and $b\in I$, why is $ab \in I$? And, does this depend on the particular choice of generators of $I$ as a $\mathbb{Z}$-module or would it be true for the $\mathbb{Z}$-span of any set from $R[G]$?
Thanks in advance.