Linear systems over localizations of rings

We describe a method for solving linear systems over the localization of a commutative ring $R$ at a multiplicatively closed subset $S$ that works under the following hypotheses: the ring $R$ is coherent, i.e., we can compute finite generating sets of row syzygies of matrices over $R$, and there is an algorithm that decides for any given finitely generated ideal $I \subseteq R$ the existence of an element $r$ in $S \cap I$ and in the affirmative case computes $r$ as a concrete linear combination of the generators of $I$.