Exercise 13, § 1

Let E be an A-module and B the ring $\mathrm{End}_A E$. Show that for an element $u \in B$ the following three conditions are equivalent: (1) $Bu$ is a direct factor of the left B-module $B_s$; (2) $uB$ is a direct factor of the right B-module $B_d$; (3) $\mathrm{Ker}(u)$ and $\mathrm{Im}(u)$ are direct factors of the A-module E. (Observe that every left ideal which is a direct factor of $B_s$ is of the form $Bp$, where $p$ is a projector in E; cf. I, § 8, Exercise 12 and use II, § 1, no. 9, Corollaries 1 and 2 to Proposition 15.)