Definition 1

Let $A$ be a ring, $E$ a right $A$-module and $M$ a left $A$-module. $E$ is said to be flat for $M$ (or $M$-flat) if, for every left $A$-module $M'$ and every injective homomorphism $v : M' \to M$, the homomorphism $l_E \otimes v : E \otimes_A M' \to E \otimes_A M$ is injective.

For any right $A$-module $N$, the notion of an $N$-flat left module is defined similarly. To say that a right $A$-module $E$ is flat for a left $A$-module $M$ is equivalent to saying that $E$, considered as a left $A^0$-module (recall that $A^0$ denotes the opposite ring to $A$), is flat for the right $A^0$-module $M$.