Introduction to Commutative Algebra
《Introduction to Commutative Algebra》 · 第 29 页 · Chapter 2「Modules」
The functor $T_N:M\mapsto M\otimes_A N$ on the category of $A$-modules and homomorphisms is therefore not in general exact. If $T_N$ is exact, that is to say if tensoring with $N$ transforms all exact sequences into exact sequences, then $N$ is said to be a flat $A$-module.