TITLE: Is a submodule of a flat module flat?
QUESTION [8 upvotes]: This is probably obvious but I am getting stuck thinking about it:
Let $A$ be a commutative ring with unity, $M$ a flat $A$-module, and $N\subset M$ a submodule.

Is $N$ necessarily flat over $A$?

I haven't found a simple counterexample, but I also haven't found more than a vague intuitive reason to think so. Thanks in advance.

REPLY [4 votes]: The property that submodules of flat modules are flat is equivalent to being locally a valuation domain (or in homological terms, having weak global dimension at most 1), see https://stacks.math.columbia.edu/tag/092S for reference. To expand on The L's answer, this is in turn equivalent to having flat ideals.  Since being reduced is a local property, this shows that any ring with nilpotents will serve as a counter example to your question,
and in fact will have non-flat principal ideals.