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# 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 &supe; Ra2 &supe; Ra3 &supe; &bull; &bull; &bull; (for a &isin; R) stabilize. Show that R is Dedekind-finite, and every non right-0-divisor in R is a unit.
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Expert's answer
2012-11-19T07:47:33-0500
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 &alpha; : RR &rarr; RR be an injective R-endomorphism.Then &alpha; is rightmultiplication by a : = &alpha;(1), and &alpha;i is right multiplication by ai. Therefore, im(&alpha;i) = Rai. Since the chain Ra &supe; Ra2 &supe; &middot; &middot; &middot; stabilizes,we have that &alpha; is an isomorphism.

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