Flat and faithfully flat modules and ring homomorphisms

Let $R$ be a commutative ring and let $M$ be an $R$-module. The module $M$ is flat if the functor $-{\otimes }_{R}M$ preserves exact sequences: whenever $A\to B\to C$ is exact, so is

$A{\otimes }_{R}M⟶B{\otimes }_{R}M⟶C{\otimes }_{R}M\mathrm{.}$

Since tensoring is always right exact (Tensoring is right exact, Exact sequences and short exact sequences of modules), the definition asks for the remaining left-hand exactness. Its equivalent formulation as preservation of injections is proved separately rather than built into the definition.

The module $M$ is faithfully flat if a sequence of $R$-modules is exact exactly when its tensor with $M$ is exact.

For a unital ring homomorphism $f:R\to S$ (Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$) between commutative rings, $S$ is an $R$-module by $r\cdot s=f(r)s$. The map $f$ is flat, respectively faithfully flat, when this $R$-module is flat, respectively faithfully flat.