Exercise 24, § 9

Let A be a ring and $\mathfrak{r}$ its radical. Prove that the following properties are equivalent:

(i) Every finitely generated left A-module admits a projective cover.

(i’) Every finitely generated right A-module admits a projective cover.

(ii) The ring $A/\mathfrak{r}$ is semisimple, and every idempotent in $A/\mathfrak{r}$ is the class of an idempotent in A.

(To prove (i)$\Rightarrow$(ii), observe that every decomposition $A/\mathfrak{r}=\mathfrak{a}_1\oplus \mathfrak{a}_2$ into a direct sum of ideals lifts to a decomposition $A = P_1\oplus P_2$, where $P_i$ is a projective cover of $\mathfrak{a}_i$. Show that, moreover, (ii) implies that every $A/\mathfrak{r}$-module is of the form $(A/\mathfrak{r})\otimes_AP$, where P is a projective A-module.)