Let $S$ be a (motivic symmetric) ring spectrum (more generally, one can possibly consider a ring object in a symmetric stable model category); let $R$ be a flat associative commutative unital algebra over the integers. I would like to consider a certain $R$-linear triangulated category of ("highly structured"?) modules over $S$. What is the "optimal" construction for it?

Actually, I am mostly interested in the case when $R$ is a localization of $\mathbb{Z}$ (whereas $S$ represents $K$-theory); so it is probably ok for my purposes to localize the corresponding category of $S$-modules. Yet I wonder whether one can construct a certain ring spectrum $S\otimes R$ instead?