nLab
flat module

Given a ring (or $k$-algebra) $A$, a left $A$-module is flat if tensoring with $A$ as a functor from left $A$-modules to left $A$-modules is exact functor (sends short exact sequences to short exact sequences). Note that tensoring with $A$ is automatically right exact, so it's equivalent to require that tensoring with $A$ be left exact. Since Mod has is finitely complete, this is also eqivalent to requiring that tensoring with $A$ be a flat functor.

More explicitly, a left $A$-module $M$ is flat if and only if “everything (that happens in $M$) happens for a reason (in $A$)” — that is, if whenever some identity ${\sum }_i{a}_i{m}_i=0$ holds in $M$, we can write $m_i={\sum }_j{b}_{ij}{n}_j$ for each $i$, with coefficients such that ${\sum }_i{a}_i{b}_{ij}=0$ for each $j$. This observation (Wraith, Blass) can be put into the more general context of modelling geometric theories by geometric morphisms from their classifying toposes, or equivalently, certain flat functors from sites for such topoi.

In the case of an abelian category, the notions of exact and flat functors coincide. The ring and/or the module above may be taken nonunital.