\begin{definition}[Definition:Bilinear Mapping/Non-Commutative Ring]
Let $R$ and $S$ be rings.
Let $M$ be a right $R$-module.
Let $N$ be a left $S$-module.
Let $T$ be an $\tuple {R, S}$-bimodule.
A '''bilinear mapping''' $f: M \times N \to T$ is a mapping which satisfies:
{{begin-eqn}}
{{eqn | n = 1
      | q = \forall r \in R: \forall s \in S: \forall m \in M: \forall n \in N
      | l = \map f {r m, s n}
      | r = r \cdot \map f {m, n} \cdot s
}}
{{eqn | n = 2
      | q = \forall m_1, m_2 \in M : \forall n \in N
      | l = \map f {m_1 + m_2, n}
      | r = \map f {m_1, n} + \map f {m_2, n}
}}
{{eqn | n = 3
      | q = \forall m \in M : \forall n_1, n_2 \in N
      | l = \map f {m, n_1 + n_2}
      | r = \map f {m, n_1} + \map f {m, n_2}
}}
{{end-eqn}}
\end{definition}