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 *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*, *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.

suppose

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