Multilinear Maps

Definition: Multilinear Maps

Let $M_1,\dots ,M_n$ be a collection of $R$-modules and let $N$ be an $R$-module. Then, the map $f:{⨁}_{i=1}^n{M}_i\to N$ is said to be multilinear if it is linear in each entry. That is, $f(m_1,\dots ,rx+y,\dots ,m_n)=rf(m_1,\dots ,x,\dots ,m_n)+f(m_1,\dots ,y,\dots ,m_n)$

Using multilinear maps allows us to apply the universal property of the tensor product to tensor products of more than two $R$-modules.