Let $g,g_i\,{\in}\,G$ and $r,r_i\,{\in}\,R$
If $(G,+_g)$ is an Abelian group and $(R,+_r,*_r)$ is a ring, then we call objects $(G,+_g,{\times}_m)$ left $R$-modules iff ${\times}_m:R{\times}G{\rightarrow}G$, ${\times}_m$ is distributive and associative, and right $R$-modules iff ${\times}_m:G{\times}R{\rightarrow}G$ with distributive and associative ${\times}_m$.
In left and right modules, $g{\times}_mr$ isn't simply not necessarily equal to $r{\times}_mg$ - in both cases, one of these is undefined, it doesn't exist.
The only exception is when $G=R$ and $+_G=+_R$, but I'm pretty sure there are bimodules other than ones in which these properties are met.
How is ${\times}_m$ defined in bimodules?
Are there simply two different functions, ${\times}_{m_L}$ and ${\times}_{m_R}$, instead of a single one like in left and right modules, or is there indeed a single ${\times}_m$ in bimodules?