Definition:Bilinear Mapping

This page has been identified as a candidate for refactoring of medium complexity. In particular: Equivalent definitions to be extracted and put into separate pages Until this has been finished, please leave {{Refactor}} in the code. New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code.

Definition

Let $\struct {R, +_R, \times_R}$ be a commutative ring.

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

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

$a_1 \mapsto a_1 \oplus a_2$ 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, $\forall a, b \in R, \forall x, y \in A_2, z \in A_3$:

$\paren {\paren {a \circ_1 x} +_1 \paren {y \circ_1 b} } \oplus z = \paren {a \circ_3 \paren {x \oplus z} } +_3 \paren {\paren {y \oplus z} \circ_3 b}$

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

$z \oplus \paren {\paren {a \circ_2 x} +_2 \paren {y \circ_2 b} } = \paren {a \circ_3 \paren {z \oplus x} } +_3 \paren {\paren {z \oplus y} \circ_3 b}$

Equivalently, this can be expressed:

$\paren {x +_1 y} \oplus z = \paren {x \oplus z} +_3 \paren {y \oplus z}$

$z \oplus \paren {x +_2 y} = \paren {z \oplus x} +_3 \paren {z \oplus y}$

$\paren {a \circ_1 x} \oplus z = a \circ_3 \paren {x \oplus z}$

$z \oplus \paren {y \circ_2 b} = \paren {z \oplus y} \circ_3 b$

If $\struct {A, +, \circ}_R = A_1 = A_2 = A_3$, the notation simplifies considerably:

$\paren {\paren {a \circ x} + \paren {b \circ y} } \oplus z = \paren {a \circ \paren {x \oplus z} } + \paren {b \circ \paren {y \oplus z} }$

$z \oplus \paren {\paren {a \circ x} + \paren {y \circ b} } = \paren {a \circ \paren {z \oplus x} } + \paren {\paren {z \oplus y} \circ b}$

or equivalently, more easily digested:

$\paren {x + y} \oplus z = \paren {x \oplus z} + \paren {y \oplus z}$

$z \oplus \paren {x + y} = \paren {z \oplus x} + \paren {z \oplus y}$

$\paren {a \circ x} \oplus z = a \circ \paren {x \oplus z}$

$z \oplus \paren {y \circ b} = \paren {z \oplus y} \circ b$

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 $\tuple {R, S}$-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

Sources

There are no source works cited for this page. Source citations are highly desirable, and mandatory for all definition pages. Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page.