Algebras and modules in monoidal model categories

with Stefan Schwede

ABSTRACT. One of the main motivations for defining a closed symmetric monoidal category of spectra is to study the associated categories of ring, algebra and module spectra. For all of the standard tools of homotopy theory to apply, Quillen model category structures must be constructed on these associated categories. In this paper we give general sufficient conditions for producing Quillen model category structures on categories of rings, algebras and modules. These conditions were created with the particular cases of symmetric spectra and $\Gamma$-spaces in mind but also apply to many other model categories with symmetric monoidal products. For example, they also produce model categories on differential graded rings, algebras and modules.