Answer to Question #16533 in Abstract Algebra for Hym@n B@ss

Question

What of the following is true for any ring R?
(a) If an is a unit in R, then a is a unit in R.
(b) If a is left-invertible and not a right 0-divisor, then a is a unit in R.
(c) If R is a domain, then R is Dedekind-finite.

Answer 1

All these are true:

For (a), note that if anc = can = 1, then a has a right inverse a^(n−1)c and a left inverse ca^(n−1), so a ∈U(R).

For (b), say ba = 1. Then (ab − 1)a = a − a = 0. If a is not a right 0-divisor, then ab = 1 and so a ∈U(R).

(c) follows immediately from (b).