# Answer to Question #16534 in Abstract Algebra for Hym@n B@ss

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.

(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

(2) Write

*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.*Need a fast expert's response?

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