Exercise 19, § 14

Let D be a field, Z its center, and V a vector space over D. Identify D with the subfield of homotheties of V. Let A be a dense subring of End$_D(V)$ (VIII, p. 129, Exercise 3) containing Z. Let E be a subfield of D containing Z and F its commutant in D. Prove that the Z-algebra $A\otimes_ZE$ is isomorphic to a dense subalgebra of End$_F(V)$ (use Exercise 14 of VIII, p. 227, and consider the commutant of AE in End$_Z(V)$). This is the case when E is a maximal subfield of D.