Exercise 16, § 5

Let D be a field, V a vector space of dimension $\geqslant 2$ over D, and A a subring of End$_D(V)$.

a) Suppose that for every system of four elements $(x, x', y, y')$, where $x$ and $x'$ are linearly independent, there exists an element $u$ of A such that $u(x) =y$ and $u(x') =y'$. Prove that the commutant of the A-module V is D (identified with the ring of homotheties of V) and that the ring of homotheties of this module is dense in its bicommutant.

b) Give an example of a ring $A\subset$ End$_D(V)$ such that V is a simple A-module but that its commutant is distinct from D (consider a field E containing D, a vector space V over E, and the ring A = End$_E(V)$).