Exercise 14, § 2

(a) Let A be a left Noetherian ring (not necessarily commutative) and $a$ a left ideal of A. Suppose that there exists a left ideal $b \neq 0$ such that $a \cap b = 0$. Show that every element $a \in a$ is a right divisor of $0$. (Let $b \neq 0$ be an element of $b$ and let $a_1 = Ab + Aba + \ldots + Aba^n$; consider the smallest integer $n$ such that $a_n = a_{n+1}$.

(b) Let $E$ be an infinite-dimensional vector space over a field $K$ and $A$ the ring $\mathrm{End}_K(E)$. Give an example of two elements $u, v$ of $A$ such that neither $u$ nor $v$ is a right divisor of $0$ and $Au \cap Av = 0$.