# Definition:Bilinear Mapping/Non-Commutative Ring

## Definition

Let $R$ and $S$ be rings.

Let $M$ be a right $R$-module.

Let $N$ be a left $S$-module.

Let $T$ be an $(R,S)$-bimodule.

A bilinear mapping $f: M \times N \to T$ is a mapping which satisfies:

$\forall r \in R: \forall s \in S: \forall m \in M: \forall N \in N$:

$f \left({r m, s n}\right) = r \cdot f\left({m, n}\right) \cdot s$

$\forall m_1, m_2 \in M : \forall n \in N$:

$f \left({m_1 + m_2, n}\right) = f \left({m_1, n}\right) + f \left({m_2, n}\right)$

$\forall m \in M : \forall n_1, n_2 \in N$:

$f \left({m, n_1 + n_2}\right) = f \left({m, n_1}\right) + f \left({m, n_2}\right)$