Thinking of DR as a new stable homotopy category, where R is a commutative S-algebra, we can realize the action of an element x ∈ Rn on an R-module M as a map of R-modules x : ΣnM −→ M. We define M/xM to be the cofiber of x, and we define the localization M[x −1 ] to be the telescope of a countable iterate of desuspensions of x, starting with M −→ Σ −nM. By iteration, we can construct quotients by sequences of elements and localizations at sequences of elements. We define R-ring spectra, associative R-ring spectra, and commutative R-ring spectra in the homotopical sense, with products A ∧R A −→ A defined via maps in the derived category DR, and it turns out to be quite simple to study when quotients and localizations of R-ring spectra are again R-ring spectra

We shall construct Bousfield localizations of R-modules at a given R-module E. In principle, this is a derived category notion, but we shall obtain precise point-set level constructions. Using different point-set level constructions, we shall prove that the Bousfield localizations of R-algebras can be constructed to be R-algebras and the Bousfield localizations of commutative R-modules can be constructed to be commutative R-algebras. In particular, the localization RE of R at E is a commutative R-algebra, and we shall see that the category of RE-modules plays an intrinsically central role in the study of Bousfield localizations.

As a very special case, this theory will imply that the spectra KO and KU that represent real and complex periodic K-theory can be constructed as commutative algebras over the S-algebras ko and ku that represent real and complex connective K-theory. Therefore KO and KU are commutative S-algebras, as had long been conjectured in the earlier context of E∞ ring spectra. Again, it is far simpler to prove the sharper ko and ku-algebra statements than to construct S-algebra structures directly.