Corollary

With each class P of projective A-modules, associate the class $\varphi (P)$ of the module $P/\mathfrak{r}P$ over the ring $A/\mathfrak{r}$. Every class of modules (resp. of finitely generated modules) over the ring $A/\mathfrak{r}$ is of the form $\varphi (P)$ for a unique class P of projective A-modules (resp. of finitely generated projective A-modules).

Let A be a left Artinian ring and $\mathfrak{r}$ its radical. Denote the set of classes of simple A-modules by $\mathscr{S}$; it is finite (VIII, p. 51). By Proposition 2 of VIII, p. 174, it can be canonically identified with the set of classes of simple modules over the semisimple ring $A/\mathfrak{r}$. For every $\lambda \in \mathscr{S}$, let $S_{\lambda}$ be a simple A-module of class $\lambda$, and choose a projective cover $(P_{\lambda}, u_{\lambda})$ of $S_{\lambda}$ (VIII, p. 175, Proposition 4). By Proposition 5, the A-module $P_{\lambda}$ is projective and indecomposable, and $u_{\lambda}$ defines an isomorphism from $P_{\lambda}/\mathfrak{r}P_{\lambda}$ to $S_{\lambda}$. Moreover, if P is a projective and indecomposable A-module, then there exists a unique $\lambda \in \mathscr{S}$ such that P is isomorphic to $P_{\lambda}:$ it is the unique $\lambda \in \mathscr{S}$ such that $P/\mathfrak{r}P$ is isomorphic to $S_{\lambda}$.