## Definition

Let $M, N, P$ be abelian groups.

Let $M \times N$ be the cartesian product.

A biadditive mapping $f : M \times N \to P$ is a mapping such that:

$\forall m_1, m_2 \in M : \forall n \in N : f \left({m_1 + m_2, n}\right) = f \left({m_1, n}\right) + f \left({m_2, n}\right)$
$\forall m \in M : \forall n_1, n_2 \in N : f \left({m, n_1 + n_2}\right) = f \left({m, n_1}\right) + f \left({m, n_2}\right)$

## Also known as

A biadditive mapping is also known as a $\Z$-bilinear mapping. See Correspondence between Abelian Groups and Z-Modules.

## Sources

• 1998: N. Bourbaki: Algebra I Chapter II. Linear Algebra $\S3$. Tensor Products. 1. Tensor product of two modules