Rings without infinite sets of noncentral orthogonal idempotents

Let $A$ be a ring without infinite sets of noncentral orthogonal idempotents. $A$ is an exchange ring if and only if all Pierce stalks of $A$ are semiperfect rings. All $A$-modules are $I_0$-modules if and only if either $A$ is a right semi-Artinian ring in which every proper right ideal is the intersection of maximal right ideals or $A/\operatorname{SI}(A_A)$ is an Artinian serial ring such that the square of the Jacobson radical of $A/\operatorname{SI}(A_A)$ is equal to zero.