Answer to Question #17649 in Abstract Algebra for Hym@n B@ss

Question

Let R be a ring in which all descending chains Ra ⊇ Ra2 ⊇ Ra3 ⊇ • • • (for a ∈ R) stabilize. Show that R is Dedekind-finite, and every non right-0-divisor in R is a unit.

Answer 1

We know that the left regular moduleRR is cohopfian iff every non right-0-divisor in R is a unit. Inthis case, show that RR is also hopfian. Then, it is sufficient to showthat RR is cohopfian. Let α : RR → RR be an injective R-endomorphism.Then α is rightmultiplication by a : = α(1), and αi is right multiplication by ai. Therefore, im(αi) = Rai. Since the chain Ra ⊇ Ra2 ⊇ · · · stabilizes,we have that α is an isomorphism.