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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