# “Extending scalars” for (motivic) ring spectra and for modules over them: are the corresponding Moore spectra highly structured ring objects?

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).

• What's wrong with simply taking the (left) derived tensor product of S and R? – Dmitri Pavlov Jun 2 '15 at 13:55
• Why is it a (highly structured) ring spectrum? This could be obviously true (I am not a specialist); yet is there any reference for this fact? – Mikhail Bondarko Jun 2 '15 at 19:50
• The construction works in any monoidal model category, of which motivic spectra are an example. For example, see Theorem 4.4 in Schwede and Shipley's “Algebras and modules in monoidal model categories”. – Dmitri Pavlov Jun 3 '15 at 7:12
• You are welcome. Feel free to ask any further questions. – Dmitri Pavlov Jun 3 '15 at 17:56
• Is your $K$-theory spectrum periodic? – Sean Tilson Jun 4 '15 at 8:58