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