Lemma 4

There exists on the additive group $\mathscr{R}(\mathfrak{a})$ a unique multiplication distributive over addition such that $[E][F] = [E\otimes F]$ for all finite dimensional $\mathfrak{a}$-modules E, F. In this way $\mathscr{R}(\mathfrak{a})$ is given the structure of a commutative ring. The class of the trivial $\mathfrak{a}$-module of dimension 1 is the unit element of this ring.

The uniqueness is clear. There exists a commutative multiplication on $\mathscr{R}(\mathfrak{a}) =\mathbf{Z}^{(\mathfrak{S}_{\mathfrak{a}})}$ that is distributive over addition and such that $[E][F] = [E\otimes F]$ for all $E,F\in \mathfrak{S}_{\mathfrak{a}}$. Let $E_1,E_2$ be finite dimensional $\mathfrak{a}$-modules, $l_1$ and $l_2$ their lengths; we show that $[E_1][E_2] = [E_1\otimes E_2]$ by induction on $l_1+l_2$. This is clear if $l_1+l_2\leq 2$. On the other hand, let $F_1$ be a submodule of $E_1$ distinct from 0 and $E_1$. Then

$[F_1][E_2] = [F_1\otimes E_2]$ and $[E_1/F_1][E_2] = [(E_1/F_1)\otimes E_2]$ by the induction hypothesis. On the other hand, $(E_1\otimes E_2)/(F_1\otimes E_2)$ is isomorphic to $(E_1/F_1)\otimes E_2$. Hence

which proves our assertion. It follows immediately that the multiplication defined above is associative, so $\mathscr{R}(\mathfrak{a})$ has the structure of a commutative ring. Finally, it is clear that the class of the trivial $\mathfrak{a}$-module of dimension 1 is the unit element of this ring.