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 $\tuple {R, S}$-bimodule.

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

\(\text {(1)}: \quad\)

\(\ds \forall r \in R: \forall s \in S: \forall m \in M: \forall n \in N: \, \)

\(\ds \map f {r m, s n}\)

\(=\)

\(\ds r \cdot \map f {m, n} \cdot s\)

\(\text {(2)}: \quad\)

\(\ds \forall m_1, m_2 \in M : \forall n \in N: \, \)

\(\ds \map f {m_1 + m_2, n}\)

\(=\)

\(\ds \map f {m_1, n} + \map f {m_2, n}\)

\(\text {(3)}: \quad\)

\(\ds \forall m \in M : \forall n_1, n_2 \in N: \, \)

\(\ds \map f {m, n_1 + n_2}\)

\(=\)

\(\ds \map f {m, n_1} + \map f {m, n_2}\)

Also see

• Definition:Bilinear Form