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

Question

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

Answer 1

The first statement is clear sinceinjective endomorphisms of 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, and
suppose ab = 1. Then xa = 0 =⇒ xab = 0 =⇒ x = 0, soa is not a right-0-divisor. It follows that a ∈ U(R), so we have shown that R is Dedekind-finite. Then RRis hopfian.