= Solution
{c}
The <modular representation ring> $R_k(G)$ is the abelian group generated by isomorphism classes $[V]$ of finite-dimensional $kG$-<modules>, with relations
$$
[V]=[U]+[W]\quad\text{whenever }0\to U\to V\to W\to0\text{ is exact}.
$$
The <Jordan–Hölder theorem> says that it is a free abelian group with basis the classes of the <simple modules>, and $[V]=\sum_S[V:S][S]$. To see independence, composition multiplicity for each fixed simple $S$ is an additive map to $\mathbb Z$ and reads off its basis coefficient.
Define $[V][W]=[V\otimes_k W]$, using the diagonal action in the <tensor product of group representations>. Tensoring over a <field> is exact in either argument, so multiplication respects the defining relations. Associativity and the symmetry $v\otimes w\mapsto w\otimes v$ give a <commutative ring> with identity $[k]$, the <trivial representation>. Thus $\boxed{R_k(G)\text{ is a commutative ring, freely based on simple modules}}$.
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