Finitely generated iff direct limits of subobjects are bounded by subobjects

Question

In the article The Dual of the Notion of "Finitely Generated", the following proposition appears.

Proposition. A module $M$ is finitely generated if and only if every direct system of proper submodules of $M$ is bounded above by a proper submodule of $M$.

In which algebraic categories does this generally hold? (A model is finitely generated if it is the quotient of a free model over a finite set.)

Answer 1

The quoted characterisation of finite generation has the unpleasant feature of referring to proper subobjects. Here is a better one:

An object $M$ is finitely generated if every directed system of subobjects of $M$ whose least upper bound is $M$ already contains $M$.

If you think for a little while you will realise that this is essentially the definition of a compact topological space if you replace "subobject" with "open subspace".

In the context of categories of (one sorted, finitary) algebraic structures, this notion of finitely generated object coincides with the classical one: just think concretely in terms of generators and unions.