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: $$\forall \left({g_1, g_2}\right) \in G_1 \times G_2$$:
 * $$g_1 \mapsto g_1 \oplus g_2$$ is a linear transformation from $$G_1$$ to $$G_3$$;
 * $$g_2 \mapsto g_1 \oplus g_2$$ is a linear transformation from $$G_2$$ to $$G_3$$.

Then $$\oplus$$ is a bilinear mapping.

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


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

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


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