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

In the case where $S$ is just the unit spectrum, this question appears to be equivalent to the following one: does the Moore spectrum corresponding to $R$ admits the structure of a highly structured ring object? I think this should be easy; yet I wasn't able to find any references even for "the usual" $SH$ (in particular, in this community the case $R=\mathbb{Z}/p^i$ was considered, and I am not interested in it).