Intersection of submodules

Question

I have question regarding intersection of submodules.

Could anyone give example of a commutative ring $R$ with identity and an $R$-module $M$ such that $$IM\cap JM\nsubseteq (I\cap J)M$$ for some ideals $I$ and $J$ of $R$?

Thank you.

Answer 1

The equality is ensured by the flatness of $M$. This leads us to consider "classical" examples of non-flat modules such as: $R=k[x^2,x^3]$, $M=k[x]$, $I=x^2R$ and $J=x^3R$.