# Finitely generated iff direct limits of subobjects are bounded by subobjects

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.)

• It is always true. Think very concretely in terms of generators and unions of subsets. – Zhen Lin Oct 12 '15 at 12:12

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