# Definition:Multilinear Mapping

## Definition

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

Let $\left({A_1, +_1, \circ_1}\right)_R, \left({A_2, +_2, \circ_2}\right)_R, \ldots, \left({A_n, +_n, \circ_n}\right)_R, \left({A_{n+1}, +_{n+1}, \circ_{n+1}}\right)_R$ be $R$-modules.

Let $\oplus: A_1 \times A_2 \times \cdots \times A_n \to A_{n+1}$ be a multiary operator with the property that: $\forall \left({a_1, a_2, \ldots, a_n}\right) \in A_1 \times A_2 \times \cdots \times A_n$:

• $a_1 \mapsto a_1 \oplus a_2 \oplus \cdots \oplus a_n$ is a linear transformation from $A_1$ to $A_{n+1}$
• $a_2 \mapsto a_1 \oplus a_2 \oplus \cdots \oplus a_n$ is a linear transformation from $A_2$ to $A_{n+1}$
• $\vdots$
• $a_n \mapsto a_1 \oplus a_2 \oplus \cdots \oplus a_n$ is a linear transformation from $A_n$ to $A_{n+1}$

Then $\oplus$ is a multilinear mapping.