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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