Question

Show that the left regular module R is hopfian iff R is Dedekind - finite.

Answer 1

Suppose ${}_R R$ is hopfian, and suppose $ab = 1$ . Then $x \to xb$ defines a surjective endomorphism $\alpha$ of ${}_R R$ . Therefore, $\alpha$ is an automorphism. Since $\alpha(ba) = bab = b = \alpha(1)$ , we must have $ba = 1$ , so $R$ is Dedekind-finite. The converse is proved by reversing this argument.