Is a finitely generated module over a Hopfian ring Hopfian?

Question

An $R$-module $M$ is called Hopfian if every $R$-epimorphism ‎$‎f :M‎\longrightarrow ‎M$‎ is an automorphism.

Suppose that $R$ is a Hopfian ring and $M$ is a finitely generated $R$-module. Is $M$ Hopfian?

Thank you for your help.

Answer 1

I think by "Hopfian ring" you mean that $R_R$ is a hopfian module. (But if you meant "Hopfian as a ring" I will need to delete this.)

What you have called a Hopfian ring is more commonly called a Dedekind finite ring or directly finite ring.

The answer is no: There exists a ring $R$ and an integer $n$ such that $R$ is Dedekind finite, and yet $M_n(R)$ is not Dedekind finite. Since $M_n(R)\cong End(R^n_R)$, you see this means that there is a surjective $R$-linear map from $R^n\to R^n$ that isn't injective.