Definition:R-Balanced Mapping

Definition
Let $R$ be a ring.

Let $M$ be a right $R$-module and $N$ be a left $R$-module.

Let $M \times N$ be their cartesian product.

Let $P$ be an abelian group.

An $R$-balanced mapping $f : M \times N \to P$ is a biadditive mapping with:
 * $\forall m \in M : \forall n \in N : \forall r \in R : f(m \cdot r, n) = f(m, r \cdot n)$

Also see

 * Definition:Bilinear Mapping