Morita equivalence preserves finitely generated modules

Question

For the proof of Morita theory the author of the proof I'm reading uses the fact that Morita equivalence preserves finitely generated Modules. I can't see why this is true. Any help will be appreciated. Thanks!

Answer 1

An equivalence preserves every categorical property, so it suffices to show that finitely generated modules can be characterized by such a property.

But such a property exists : let $M$ be a finitely generated module and $(U_i)_{i\in I}$ a family of modules, together with an epimorphism $f: \displaystyle\bigoplus_{i\in I}U_i \to M$. Then there exists a finite $F\subset I$ such that $f\circ i$ is also an epimorphism, where $i: \displaystyle\bigoplus_{i\in F}U_i \to \displaystyle\bigoplus_{i\in I}U_i $ is the canonical injection (that can be defined in terms of a universal property). That's an easy exercise (knowing that epimorphisms in $R-Mod$ are exactly surjective maps)

Conversely, if $M$ satisfies this property, then putting $I=M$ and $U_i =R$, you get that $M$ is finitely generated.

Now if $F: R-Mod \to S-Mod$ is an equivalence and $M$ is finitely generated, it satisfies this property, and thus so does $FM$ (because $F$ preserves epimorphisms, coproducts, canonical injections - those are, again, easy exercises), and thus $FM$ is finitely generated. This proves the claim.

Note that any categorical characterization of f.g. modules will yield a proof of this fact. Similarly, finitely presented modules can be described by such a characterization, and so are also preserved by equivalences (they are the compact objects in $R-Mod$)