Exercise 14, § 1

Let V be a vector space of countable infinite dimension over a field K. The set $\mathscr{I}$ of endomorphisms of V of finite rank is a two-sided ideal of the ring B = End$_K(V)$. Prove that the only two-sided ideals of the ring $A = B/\mathscr{I}$ are $\{0\}$ and A, but that A is neither left nor right Noetherian.