Exercise 12, § 9

a) A ring A is without radical if and only if it is isomorphic to a subring C of a product $\prod_{i\in I}B_i$ of primitive rings such that pr$_i(C) = B_i$ for every $i\in I$ (consider the set of classes of simple A-modules).

b) Let V be an infinite-dimensional vector space over a field D, let B be the ring End$_D$(V), and let $\mathfrak{s}$ be its socle (cf. VIII, p. 130, Exercise 8). Let C be the subring of $B\times B$ generated by $\mathfrak{s}\times \mathfrak{s}$ and the unit element. Prove that C is without radical but is not isomorphic to a product of primitive rings.