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

Question #17649

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.

Expert's answer

We know that the left regular module

*RR*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*α**i*is right multiplication by*ai*. Therefore, im(*α**i*) =*Rai*. Since the chain*Ra**⊇**Ra*2*⊇**· · ·*stabilizes,we have that*α*is an isomorphism.
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