Localization commutes with tensor products ($T^{-1}(M{\otimes }_{A}N)\cong T^{-1}M{\otimes }_{T^{-1}A}{T}^{-1}N$)

For a multiplicatively closed set $T$ in a commutative ring $A$, the map sends $(m\otimes n)/t$ to $(m/t)\otimes (n/1)$. An inverse sends $(m/s)\otimes (n/t)$ to $(m\otimes n)/(st)$. The fraction relations and the tensor product of modules balancing relation make these maps well-defined and mutually inverse.