# Hopfian modules and equivalence of categories of modules

by uno   Last Updated May 16, 2019 04:20 AM

For a ring with unity (not necessarily commutative) $$R$$, let $$R$$-$$Mod$$ denote the category of left $$R$$-modules.

Let $$R,S$$ be two rings with unity and $$T: R$$-Mod $$\to S$$-Mod be an equivalence of categories ($$T$$ be co-variant). Is it true that $$M$$ is a Hopfian $$R$$-module if and only if $$T(M)$$ is an Hopfian $$S$$-module ?

In general, if $$R$$-Mod and $$S$$-Mod are equivalent as categories, then do we have a one-to-one correspondence between Hopfian $$R$$-modules and Hopfian $$S$$-modules ?

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