Proposition 5

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

a) Let P be a projective A-module. Denote the canonical mapping from P to $P/\mathfrak{r}P$ by $u$. Then $(P, u)$ is a projective cover of $P/\mathfrak{r}P$. In particular, the A-module P is finitely generated if and only if $P/\mathfrak{r}P$ is.

b) Let M be a module over the ring $A/\mathfrak{r}$. Then M, viewed as an A-module, has a projective cover. If $(P, u)$ is such a projective cover, then when passing to the quotient, $u$ induces an isomorphism from $P/\mathfrak{r}P$ to M. Moreover, P is indecomposable if and only if M is simple.

c) Let M and $M'$ be modules over the ring $A/\mathfrak{r}$. Let $(P, u)$ and $(P', u')$ be projective covers of the A-modules M and $M'$. Then M and $M'$ are isomorphic if and only if P and $P'$ are.

The radical $\mathfrak{r}$ of A is a nilpotent two-sided ideal (VIII, p. 173, Proposition 1). Assertion a) therefore follows from Corollary 1 of VIII, p. 163 and Remark 2 of VIII, p. 161.

Let us prove assertion b). The existence of a projective cover $(P, u)$ of M follows from Proposition 4 and the isomorphism from $P/\mathfrak{r}P$ to M of Proposition 9 of VIII, p. 162. Since the ring $A/\mathfrak{r}$ is semisimple, a module over this ring is indecomposable if and only if it is simple. The last assertion then follows from Corollary 2 of VIII, p. 160.

Let us prove c). Let $f$ be an isomorphism from P to $P'$; it induces an isomorphism $\overline{f}$ from $P/\mathfrak{r}P$ to $P'/\mathfrak{r}P'$. By b), these quotients are isomorphic to M and $M'$, respectively. Therefore, M is isomorphic to $M'$. Conversely, by Proposition 8 of VIII, p. 161, every isomorphism $f$ from M to $M'$ lifts to an isomorphism $\widetilde{f}$ from P to $P'$ such that $f\circ u=u'\circ \widetilde{f}$.