Rings, Modules, and Algebras in Stable Homotopy Theory

. Let S be the sphere spectrum. We construct an associative, commutative, and unital smash product in a complete and cocomplete category MS of "S-modules" whose derived category DS is equivalent to the classical stable homotopy category. This allows a simple and algebraically manageable definition of "S-algebras" and "commutative S-algebras" in terms of associative, or associative and commutative, products R S R \\Gamma! R. These notions are essentially equivalent to the earlier notions of A1 and E1 ring spectra, and the older notions feed naturally into the new framework to provide plentiful examples. There is an equally simple definition of R-modules in…