I've been reading this article by Voloch which gives an alternate construction of the localization $D^{-1}R$ of a ring $R$ via introducing an indeterminate $x_d$ for each $d \in D$ and quotienting by the ideal $\langle dx_d -1 \mid d \in D \rangle$. I was wondering if there was a corresponding construction for the localization of a module.
I know of the construction for $D^{-1}M$ via "fractions" or via tensoring with $S^{-1}R$. Maybe we could modify the last one...
One could construct the localized module as $$\frac{M \oplus \left( \oplus_{u \in U} M \right)}{\langle m - u\cdot m_u \rangle },$$ but it's not clear why this would be edifying, because it doesn't have the nice universal properties that the ring case does. It's probably better to think of the localization as $M \otimes_R R[U^{-1}]$, and then prove the isomorphism to $M[U^{-1}]$, which is done in many books.