Proposition 6

Let A be a left Artinian ring and $\mathfrak{r}$ its radical. Let P be a projective A-module.

a) The A-module $\overline{P}= P/\mathfrak{r}P$ is semisimple, and the A-module P is isomorphic to $\bigoplus_{\lambda\in\mathscr{S}}P^{([P:\lambda])}_{\lambda}$.

b) Let $(Q_i)_{i\in I}$ be a family of indecomposable projective submodules of P with direct sum P. Then for every $\lambda \in \mathscr{S}$, the cardinal of the set $I(\lambda )$ of $i\in I$ such that $Q_i$ is isomorphic to $P_{\lambda}$ is equal to $[P :\lambda ]$.

The fact that $\overline{P}$ is semisimple follows from Proposition 2 (VIII, p. 174). The A-module $Q =\oplus_{\lambda\in\mathscr{S}}P^{([P:\lambda])}_{\lambda}$ is projective. Since $P_{\lambda}/\mathfrak{r}P_{\lambda}$ is isomorphic to $S_{\lambda}$, the quotient $Q/\mathfrak{r}Q$ is isomorphic to $\oplus_{\lambda\in\mathscr{S}}S^{([P:\lambda])}_{\lambda}$, that is, to $\overline{P}= P/\mathfrak{r}P$. By Proposition 5, the A-modules P and Q are isomorphic.

Suppose given a family $(Q_i)_{i\in I}$ of projective and indecomposable submodules with direct sum P. By Proposition 5, the A-module $Q_i/\mathfrak{r}Q_i$ is simple for every $i\in I$. For $\lambda \in \mathscr{S}$, denote by $I(\lambda )$ the set of $i\in I$ such that $Q_i/\mathfrak{r}Q_i$ is isomorphic to $S_{\lambda}$, hence to $P_{\lambda}/\mathfrak{r}P_{\lambda}$; it is also the set of $i\in I$ such that $Q_i$ is isomorphic to $P_{\lambda}$. Since $\overline{P}= P/\mathfrak{r}P$ is the direct sum of the family $(Q_i/\mathfrak{r}Q_i)_{i\in I}$, we have Card(I($\lambda$ )) $= [P :\lambda ]$ by Theorem 1 of VIII, p. 32.