## Balanced Product

For a ring $R$, a right $R$-module $M$, a left $R$-module $N$, and an abelian group $G$, a map $\phi:M\times N\to G$ is said to be $R$-balanced, if for all $m,m'\in M$, $n,n'\in N$, and $r\in R$ the following hold:

\begin{aligned} \phi(m,n+n')&=\phi(m,n)+\phi(m,n')\\ \phi(m+m',n)&=\phi(m,n)+\phi(m',n)\\ \phi(m\cdot r,n)&=\phi(m,r\cdot n) \end{aligned}

The first two axioms are essentially bilinearity, while the third is something like associativity.