Balanced Map of Modules

Let $M$ be an R-S-Bimodule, $N$ a S-T-Bimodule, and $H$ a R-T-Bimodule. A function $\varphi:M\times N\to H$ is said to be balanced if:

• $\varphi(m_{1}+m_{2},n)=\varphi(m_{1},n)+\varphi(m_{2},n)$
• $\varphi(m,n_{1}+n_{2})=\varphi(m,n_{1})+\varphi(m,n_{2})$
• $\varphi(m\cdot s,n)=\varphi(m,s\cdot n)$
• $\varphi(r\cdot m,n)=r\cdot\varphi(m,n)$
• $\varphi(m,n\cdot t)=\varphi(m,n)\cdot t$

Arguably, balanced maps are the "right" notion of Bilinearity, as bilinear maps are balanced maps where $S$ is taken to be the Ring of Integers. Some authors also require that $R$ and $T$ be $\mathbb{Z}$ as well.