Exercise 3, § 3

Let D be a field, V a right D-vector space, and A the endomorphism ring of V. For every line L in V, we denote by $\mathfrak{m}_L$ the set of elements $u$ of A such that $u(L) = 0$. Prove that $\mathfrak{m}_L$ is a maximal ideal of A, that we have $\mathfrak{m}_L\not=\mathfrak{m}_{L'}$ for $L\not= L'$, but that the simple A-modules $A/\mathfrak{m}_L$ are all isomorphic to the A-module V.