# Definition:Bilinear Mapping

## Definition

Let $\left({R, +_R, \times_R}\right)$ be a commutative ring.

Let $\left({A_1, +_1, \circ_1}\right)_R, \left({A_2, +_2, \circ_2}\right)_R, \left({A_3, +_3, \circ_3}\right)_R$ be $R$-modules.

Let $\oplus: A_1 \times A_2 \to A_3$ be a binary operator with the property that: $\forall \left({a_1, a_2}\right) \in A_1 \times A_2$:

$a_1 \mapsto a_1 \oplus a_2$ is a linear transformation from $A_1$ to $A_3$
$a_2 \mapsto a_1 \oplus a_2$ is a linear transformation from $A_2$ to $A_3$

Then $\oplus$ is a bilinear mapping.

That is, $\forall a, b \in R, \forall x, y \in A_2, z \in A_3$:

$\left({\left({a \circ_1 x}\right) +_1 \left({y \circ_1 b}\right)}\right) \oplus z = \left({a \circ_3 \left({x \oplus z}\right)}\right) +_3 \left({\left({y \oplus z}\right) \circ_3 b}\right)$

and for all $z \in A_1, x,y \in A_2$:

$z \oplus \left({\left({a \circ_2 x}\right) +_2 \left({y \circ_2 b}\right)}\right) = \left({a \circ_3 \left({z \oplus x}\right)}\right) +_3 \left({\left({z \oplus y}\right) \circ_3 b}\right)$

Equivalently, this can be expressed:

$\left({x +_1 y}\right) \oplus z = \left({x \oplus z}\right) +_3 \left({y \oplus z}\right)$
$z \oplus \left({x +_2 y}\right) = \left({z \oplus x}\right) +_3 \left({z \oplus y}\right)$
$\left({a \circ_1 x}\right) \oplus z = a \circ_3 \left({x \oplus z}\right)$
$z \oplus \left({y \circ_2 b}\right) = \left({z \oplus y}\right) \circ_3 b$

If $\left({A, +, \circ}\right)_R = A_1 = A_2 = A_3$, the notation simplifies considerably:

$\left({\left({a \circ x}\right) + \left({b \circ y}\right)}\right) \oplus z = \left({a \circ \left({x \oplus z}\right)}\right) + \left({b \circ \left({y \oplus z}\right)}\right)$
$z \oplus \left({\left({a \circ x}\right) + \left({y \circ b}\right)}\right) = \left({a \circ \left({z \oplus x}\right)}\right) + \left({\left({z \oplus y}\right) \circ b}\right)$

or equivalently, more easily digested:

$\left({x + y}\right) \oplus z = \left({x \oplus z}\right) + \left({y \oplus z}\right)$
$z \oplus \left({x + y}\right) = \left({z \oplus x}\right) + \left({z \oplus y}\right)$
$\left({a \circ x}\right) \oplus z = a \circ \left({x \oplus z}\right)$
$z \oplus \left({y \circ b}\right) = \left({z \oplus y}\right) \circ b$

### 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 $(R,S)$-bimodule.

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

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

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

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

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

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

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