Question #16534

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.

Expert's answer

(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.*

(2) Write

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