# Inclusion of two submodules generated by ideals

If $A$ is a commutative ring, $M$ a finitely generated $A$ module and $\mathfrak{a}M \subset \mathfrak{b}M$ with ideals $\mathfrak{a}, \mathfrak{b} \subset A$, does this imply $\mathfrak{a} \subset \mathfrak{b}$? Maybe with additional assumptions on $A$ or the ideals? Thanks and regards.

-
Not at all: take $M=0$. – Georges Elencwajg Jan 22 '12 at 13:56
Also, it's really important that you explain what means $\subset$. Is $\subseteq$ or $\subsetneq$? – emiliocba Jan 22 '12 at 14:10
Of course, how obvious, thanks. I recommend deleting this stupid question, there's nothing to learn for anyone. – Joni Jan 22 '12 at 17:21