The smash product for derived categories in stable homotopy theory

An E1E1 (or A∞A∞) ring spectrum RR has a derived category of modules DRDR. An E2E2 structure on RR endows DRDR with a monoidal product ∧R∧R. An E3E3 structure on RR endows ∧R∧R with a braiding. If the E3E3 structure extends to an E4E4 structure then the braided monoidal product ∧R∧R is symmetric monoidal.