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.

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