Example

Let us consider the case when the ring A is commutative. The mapping that sends a maximal ideal $\mathfrak{m}$ onto cl(A$/\mathfrak{m}$) is a bijection from the set of maximal ideals of A to the set $\mathscr{S}$ of classes of simple A-modules (VIII, p. 51). The inverse bijection sends $\lambda$ to its annihilator $\mathfrak{m}_{\lambda}$.

Let M be an A-module. For every $\lambda \in \mathscr{S}$, the isotypical component $M_{\lambda}$ of type $\lambda$ of M consists of the elements annihilated by $\mathfrak{m}_{\lambda}$, and we can view $M_{\lambda}$ as a vector space over the field $A/\mathfrak{m}_{\lambda}$. If M is semisimple and N is another A-module, then we can deduce a group isomorphism from Hom$_A(M,N)$ to $\prod_{\lambda\in\mathscr{S}}$ Hom$_{A/\mathfrak{m}_{\lambda}}(M_{\lambda},N_{\lambda})$ from Proposition 5.