Exercise 15, § 9

Let $V_0$ be a vector space of finite dimension $n$ over a field D, and let B be a subring of the ring End$_D(V_0)$. Set $V_k= V_0$ for $k\geqslant 1$ and $V =\oplus_{k\geqslant 0}V_k$. Consider the set A of endomorphisms $u$ of V for which there exists an integer $m$ (that depends on $u$) such that we have $u(V_k)\subset \oplus_{i\leqslant m}V_i$ for $k\leqslant m$ and $u(V_k)\subset V_k$ for $k > m$ and that for $k > m$, the endomorphism of $V_k$ obtained by restriction is independent of $k$ and belongs to B.

a) Prove that A is a primitive ring and that its socle consists of the elements $u$ of A such that $u(V_k) = 0$ for all but finitely many indices $k$ (VIII, p. 130, Exercise 8). b) Choose suitable $D,n$, and B to give an example of a primitive ring that has a quotient ring with nilpotent nonzero radical and an example of a primitive ring whose center has nonzero radical.