On Modules Over Motivic Ring Spectra

In this note, we provide an axiomatic framework that characterizes the stable $infty$-categories that are module categories over a motivic spectrum. This is done by invoking Luries $infty$-categorical version of the Barr--Beck theorem. As an application, this gives an alternative approach to Rondigs and O stvae rs theorem relating Voevodskys motives with modules over motivic cohomology, and to Garkushas extension of Rondigs and O stvae rs result to general correspondence categories, including the category of Milnor-Witt correspondences in the sense of Calm`es and Fasel. We also extend these comparison results to regular Noetherian schemes over a field (after inverting the residue characteristic), following the methods of Cisinski and Deglise.