Exercise 16, § 12

Let D be a field, V an infinite-dimensional right vector space over D, and A a dense subring of the ring End$_D(V)$ whose socle $\mathfrak{s}$ is not zero (VIII, p. 130, Exercises 8 and 9). Let K be a subfield of the center of D and B a K-algebra. Prove that if the radical of the K-algebra $D^o\otimes B$ is nonzero, then so is that of $A\otimes B$. (Let $\mathfrak{a}$ be the set of elements $\sum u_i\otimes b_i$ of $\mathfrak{s}\otimes B$ satisfying $\sum\langle u_i(x), x^*\rangle \otimes b_i\in \mathfrak{R}(D^o\otimes B)$ for all $x\in V$ and $x^*\in V^*$. Prove that $\mathfrak{a}$ is a two-sided ideal of $A\otimes B$ and, using Exercises 10, e) of VIII, p. 167 and 9 of VIII, p. 131, that it is nonzero and is contained in the radical of $A\otimes B.)$