# Answer to Question #17648 in Abstract Algebra for Hym@n B@ss

Question #17648

Let ϕ : R → S be a ring homomorphism such that S is finitely generated when it is viewed as a left R-module via ϕ. If, over R, all finitely generated left modules are hopfian (resp. cohopfian), show that the same property holds over S.

Expert's answer

Let

*f*:*M → M*be asurjective (resp. injective) endomorphism of a finitely generated left*S*-module*M*. Via*ϕ*, we may view*M*as a left*R*-module,and, since*RS*is finitely generated, so is*RM*. Viewing*f*:*M → M*as a surjective (resp. injective)*R*-homomorphism, we inferfrom the assumption on*R*that*f*is an*R*-isomorphism, andhence an*S*-isomorphism.
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