Exercise 9, § 10

Let A be an Artinian ring whose radical $\mathfrak{r}$ is contained in the center.

a) Prove that every central idempotent in $A/\mathfrak{r}$ lifts to a central idempotent in A (if $e$ is an idempotent in A with central class in $A/\mathfrak{r}$, express the fact that $e$ commutes with $xe-ex$ for every $x\in A$).

b) If $\mathfrak{r}$ is nonzero, prove that $A/\mathfrak{r}$ is commutative (observe that we have the relation $(xy-yx)z= 0$ for $x\in A,y\in A$, and $z\in \mathfrak{r}$).

c) Deduce from a) and b) that A is isomorphic to the product of a semisimple ring and a finite family of local rings $A_i$ such that the field $A_i/\mathfrak{R}(A_i)$ is commutative. d) Prove that every simple A-module is isomorphic to a minimal left ideal of A.