Question

Let a ∈ R, where R is any ring.
(1) Show that if a has a left inverse, then a is not a left 0-divisor.
(2) Show that the converse holds if a ∈ aRa.

Answer 1

(1) Say ba = 1. Then ac = 0 implies c = (ba)c = b(ac) = 0.
(2) Write a = ara, and assume a is not a left 0-divisor. Then a(1 − ra) = 0 yields ra = 1, so a has left inverse r.