http://nlab.mathforge.org/nlab/show/ring+spectrum
Nice answer on intuition for ring spectra: http://mathoverflow.net/questions/82485/dedekind-spectra
Lurie: Elliptic cohomology. Mentions three possible notions of ring spectra I think.
Introduction in Greenlees: Spectra for commutative algebraists (Homotopy theory folder). Applications: Section 6A discussion Topological HH, section 6B discusses trace maps.
Roendigs et al: Motivic strict ring models for K-theory
http://front.math.ucdavis.edu/0912.4346 Units of equivariant ring spectra, by Santhanam
http://mathoverflow.net/questions/433/e-infty-spectrum-corresponding-to-zp contains some interesting ideas on pro-objects in spectra and p-adic topologies, and module categories over spectra.
Structured Ring Spectra (London Mathematical Society Lecture Note Series) fra Mathematics by: Andrew Baker, Birgit Richter
Naumann in talk July 2009: Given a ring cohomology th, Brown rep gives a spectrum, but in general we cannot say that it is a ring spectrum, because of the possible problem of phantom maps. Hopkins have showed that in classical stable htpy th, there are no even-degree phantom maps between Landweber exact spectra, although I think there might be odd degree (example given in talk). We don’t have a general analogue of Hopkins’ result in the motivic setting, but we can treat relevant special cases.
arXiv:1001.0902 Homological dimensions of ring spectra from arXiv Front: math.AT by Mark Hovey, Keir Lockridge We define homological dimensions for S-algebras, the generalized rings that arise in algebraic topology. We compute the homological dimensions of a number of examples, and establish some basic properties. The most difficult computation is the global dimension of real K-theory KO and its connective version ko at the prime 2. We show that the global dimension of KO is 1, 2, or 3, and the global dimension of ko is 4 or 5.
arXiv:1004.0006 The smash product for derived categories in stable homotopy theory from arXiv Front: math.AT by Michael A. Mandell 1 person liked this An E_1 (or A-infinity) ring spectrum R has a derived category of modules D_R. An E_2 structure on R endows D_R with a monoidal product. An E_3 structure on R endows the monoidal product with a braiding. If the E_3 structure extends to an E_4 structure then the braided monoidal product is symmetric monoidal.
arXiv:1208.6005 Constructing model categories with prescribed fibrant objects from arXiv Front: math.CT by Alexandru E. Stanculescu We put a model category structure on the category of small categories enriched over a suitable monoidal simplicial model category. For this we use the model structure on small simplicial categories due to J. Bergner and a weak form of a recognition principle for model categories due to J.H. Smith. We give an application of this weak form of Smith’s result to left Bousfield localizations of categories of monoids in a suitable monoidal model category.
nLab page on Ring spectrum