# Answer to Question #17647 in Abstract Algebra for Hym@n B@ss

Question #17647

Show that the left regular module R is cohopfian iff every non right-0-divisor in R is a unit. In this case, show that R is also hopfian

Expert's answer

The first statement is clear sinceinjective endomorphisms of

suppose

*RR*are given by right multiplications by nonright-0-divisors, and automorphisms of*RR*are given by rightmultiplications by units. Now suppose non right-0-divisors are units, andsuppose

*ab*= 1. Then*xa*= 0 =*⇒**xab*= 0 =*⇒**x*= 0*,*so*a*is not a right-0-divisor. It follows that*a**∈**U(**R*), so we have shown that*R*is Dedekind-finite. Then*RR*is hopfian.
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