Exercise 3, § 7

Let D be a field, V a right vector space over D, and A a subring of the ring End$_D(V)$.

a) Prove that the following conditions are equivalent:

(i) For every endomorphism $u$ of V and every finite sequence $x_1, . . . , x_n$ of vectors in V, there exists an element $a$ of A satisfying $u(x_i) =a(x_i)$ for $1\leqslant i\leqslant n$.

(ii) For every pair of elements $x, y$ of V with $x\not= 0$, there exists an $a\in A$ such that $a(x) =y$.

When these conditions are satisfied, we say that the subring A of End$_D(V)$ is dense. Assume from now on that A is dense.

b) Prove that A is primitive (VIII, p. 120, Remark 2). Prove that, conversely, every primitive ring is isomorphic to a dense subring of the endomorphism ring of a vector space.

c) Prove that the nonzero elements of the center of A are injective endomorphisms. In particular, the center of a primitive ring is an integral domain.

d) Suppose that V is infinite-dimensional. For every integer $n >0$, prove that there exist a subring $A_n$ of A and a surjective homomorphism $A_n\rightarrow \mathbf{M}_n(D)$.