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

Question #16533

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.

(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.

Expert's answer

All these are true:

For (a), note that if

For (b), say

(c) follows immediately from (b).

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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