= Modular representation ring
{title2=$R_k(G)$}
The modular representation ring is the <Grothendieck group> of finite-dimensional <group representations>, with relations $[V]=[U]+[W]$ for every <short exact sequence> $0\to U\to V\to W\to0$, and multiplication $[V][W]=[V\otimes_k W]$ using the <tensor product of group representations>. The <Jordan–Hölder theorem> makes the classes of <simple modules> a free integral basis. Tensoring over a <field> is exact, so this multiplication is well defined. When $k$ is a <splitting field for finite group representations>, <Brauer characters> identify $\mathbb C\otimes_{\mathbb Z}R_k(G)$ with the algebra of complex <class functions> on <p-regular elements>. The splitting hypothesis is essential: $\mathbb F_2C_3\cong\mathbb F_2\times\mathbb F_4$ has two <simple modules>, although $C_3$ has three $2$-regular conjugacy classes.
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