Exercise 14, § 12

Let V be a vector space over a field D. Denote the endomorphism ring of the abelian group V by Ω, and identify D with a subring of Ω. Set A = End$_D(V)$.

a) Prove that A and D are linearly disjoint over their common center Z (cf. Exercise 5, d)).

b) The image of $D\otimes_ZA$ in Ω is equal to End$_Z(V)$ if and only if D has finite rank over Z. (To see that the condition is necessary, first deduce the case V = D from Exercise 5, e), and then fix an element $x\not= 0$ of V and consider the subalgebra B of End$_Z(V)$ consisting of the Z-endomorphisms $u$ of V such that $u(Dx)\subset Dx$. Observe that the equality End$_Z(V) =$ DA implies $B = D(B\cap A).)$