Flat module

An $R$-module $M$ is flat when tensoring with it preserves injections, equivalently when $-{\otimes }_{R}M$ is an exact functor.

A flat module $M$ is faithfully flat when $N{\otimes }_{R}M=0$ implies $N=0$. Every field extension is faithfully flat over its base field.

An $R$-module $M$ is flat if and only if $M_{\mathfrak{p}}$ is flat over $R_{\mathfrak{p}}$ for every prime ideal $\mathfrak{p}$, equivalently for every maximal ideal.

Over a principal ideal domain, a module is flat exactly when it is torsion-free. Hence the tensor product of two torsion-free modules over a principal ideal domain is torsion-free: both tensor functors are exact, so their composite is exact.

If an $R$-algebra $A$ is flat as an $R$-module, then for ideals $I,J\subseteq R$,Tensor the exact sequence $0\to I\cap J\to I\oplus J\to I+J\to 0$ with $A$.

A nonzero commutative ring $R$ has the property that $M{\otimes }_{R}N=0$ forces $M=0$ or $N=0$ exactly when $R$ is local with maximal ideal $\mathfrak{m}$ and $M=\mathfrak{m}M$ forces $M=0$ for every $R$-module. Reduction modulo $\mathfrak{m}$ turns a tensor product into a tensor product of vector spaces.