Remark 3

Applying the previous results to the opposite ring of A, we see, in particular, that a monogenous right A-module M is projective if and only if there exists an idempotent $e$ in A such that M is isomorphic to $eA$. Moreover, $eA$ is an indecomposable right module if and only if $e$ is indecomposable.

Now suppose that the ring A is semisimple, and denote the set of classes of simple left A-modules by $\mathscr{S}$. The A-module $A_s$ is semisimple, and every submodule of $A_s$ is a direct factor. Let $\mathfrak{a}$ be a left ideal of A. By the above, there exists an idempotent $e$ in A such that $\mathfrak{a}= Ae$, and the A-module $\mathfrak{a}$ is simple if and only if it is indecomposable, that is, if and only if $e$ is indecomposable.

Let $(\mathfrak{m}_i)_{i\in I}$ be a family of minimal left ideals of A such that we have $A_s=\oplus_{i\in I}\mathfrak{m}_i$. By Proposition 14, there exists a partition $(\varepsilon_i)_{i\in I}$ of 1 consisting of indecomposable idempotents such that $\mathfrak{m}_i= A\varepsilon_i$ for every $i\in I$.

For every $\lambda \in \mathscr{S}$, let $S_{\lambda}$ be an A-module of class $\lambda$, and let $\mathfrak{a}_{\lambda}$ be the isotypical component of type $\lambda$ of the A-module $A_s$. Since A is the direct sum of the family $(\mathfrak{a}_{\lambda})_{\lambda\in\mathscr{S}}$, there exists a partition $(e_{\lambda})_{\lambda\in\mathscr{S}}$ of 1 such that $\mathfrak{a}_{\lambda}= Ae_{\lambda}$ for every $\lambda \in \mathscr{S}$. For $\lambda \in \mathscr{S}$, denote by $I(\lambda )$ the set of indices $i\in I$ such that the simple A-module $\mathfrak{m}_i$ is of type $\lambda$. By Proposition 4, b) of VIII, p. 65, we have

The idempotent $e_{\lambda}$ is the component of 1 in $\mathfrak{a}_{\lambda}$, so $e_{\lambda}=\sum_{i\in I(\lambda)}\varepsilon_i$.