Is there a name for a module that is simultaneously Hopfian and co-Hopfian?

Question

An $R$-module $M$ is Hopfian (resp. co-Hopfian) if every surjective (resp. injective) $f:M \rightarrow M$ is an isomorphism. Do we have a name for a module $M$ that has both properties? Such a module would be a direct analogue of a finite dimensional vector space. If no name exists yet, might it be appropriate to refer to such an $M$ as a finite object in the category of $R$-modules?

Answer 1

I don't know a name for such modules (as far as I can tell from a little literature search they are just called "Hopfian and co-Hopfian"), but I think "finite object" is a profoundly inappropriate name and thinking of such modules as a generalization of finite-dimensional vector spaces is unhelpful. In particular, a Hopfian and co-Hopfian module need not be finitely generated; a simple example is that $\mathbb{Q}$ is Hopfian and co-Hopfian as a $\mathbb{Z}$-module. In fact, there even exist Hopfian and co-Hopfian $\mathbb{Z}$-modules that are uncountable.

Basically, being Hopfian and co-Hopfian doesn't have very much to do with finiteness at all. It has more to do with being "rigid" so that it's impossible to construct nontrivial injections or surjections from the module to itself. The case of vector spaces is very special since all vector spaces are quite "flexible" (they have bases which makes it very easy to define maps out of them), and so the only way to have such rigidity is to be finite-dimensional.

If you want a brief and snappy term to mean "Hopfian and co-Hopfian", I would suggest "bi-Hopfian" (though as far as I know this term has never been used before). The prefix bi- is sometimes used in categorical contexts to say that both a condition and its dual condition holds (e.g., biproducts which are both products and coproducts).

(Being Hopfian and co-Hopfian follows from certain natural finiteness properties. Specifically, any Noetherian module is Hopfian and any Artian module is co-Hopfian, so a module which is both Noetherian and Artinian (also known as a finite length module) is both Hopfian and co-Hopfian. But the converses to these statements are not true in general, because typically most submodules of a module cannot be obtained as images or kernels of an endomorphism of the module.)