Any module is a union of an increasing chain of finitely generated submodules.

Question

This was a statement on wikipedia's page on finitely generated module:

Any module is a union of an increasing chain of finitely generated submodules.

But if we look at $\mathbb{R}$ as a $\mathbb{Q}$-module, a chain contains only countably many submodules, and each submodule only contains countably many elements - how can their union by $\mathbb{R}$? Am I getting some definitions wrong?

Answer 1

Wikipedia is wrong here. I suspect that they've just misstated the fact (which is a fact) that every module is the union of a filtered set of finitely generated submodules.