Bilinear Map

Let $R,S$ be a pair of Rings, $M$ be a Left R-Module, $N$ be a Right S-Module, and $T$ be an (R,S)-Bimodule. A function $f:\langle-,-\rangle:M\times N\to T$ is a bilinear map if it is "separately linear" in each variable:

• $\langle m_{1}+m_{2},n\rangle=\langle m_{1},n\rangle+\langle m_{1},n\rangle$
• $\langle m,n_{1}+n_{2}\rangle=\langle m,n_{1}\rangle+\langle m,n_{2}\rangle$
• $\langle cm,n\rangle=c\langle m,n\rangle$
• $\langle m,nc\rangle=\langle m,n\rangle c$

More concisely, a bilinear map is simply a Linear Map $M\otimes N\to T$ from the Tensor Product of Modules.

Note that none of this requires a group structure, and can be generalized to Semirings and Semiring Bimodules.