Suppose b R a noncommutative ring with unity Suppose that ab

Suppose b R, a non-commutative ring with unity. Suppose that ab = bc = 1, that is, b has a right inverse c and a left inverse a. Prove that a =c and that b is a unit.

Solution

Let R be a non-commutative ring with unity. Let a, b R. Assume that ab = bc = 1

Now, ab = 1. Multiply both sides by c on the right hand side to obtain

(ab)c = 1. c

a (b c) = c

a (1) = c

a = c

associate and multiplicative identity properties is applied in each step

Hence b a = b c = 1,

so ab = b a = 1.

Thus b has a multiplicative inverse a. By definition of a unit, b is a unit.