symmetric monoidal (∞,1)-category of spectra
The terminology (commutative) ring spectrum refers either to a (commutative) monoid in the stable homotopy category regarded as monoidal category via the smash product of spectra, or to the richer structure of a monoid in a model structure for spectra equipped with a symmetric smash product of spectra.
In the first case a ring spectrum is a spectrum equipped with a unit and product operation, which is associative, unital (and commutative) just up to unspecified homotopy (such as in an H-space structure). In the second case the structure is much richer; and in good cases, such as for highly structured spectra, is equivalent to A-∞ ring structure (E-∞ ring structure).
To distinguish the two situations further qualification is being used. Sometimes one says homotopy ring spectrum to explicitly refer to the first case (e.g. Schwede 12, chapter II 4.1) or one says “highly structured ring spectrum” to refer explicitly to the second case. For more on this see at brave new algebra and higher algebra.
Since the concept of spectrum is the refinement of the concept of abelian group to homotopy theory/(∞,1)-category theory. The concept of ring spectrum is the corresponding generalization of the notion of (commutative) ring.
|algebra||homological algebra||higher algebra|
|abelian group||chain complex||spectrum|
Quick survey includes
Discussion of connective ring spectra as monoids with respect to a smash product on Gamma-spaces is in
A comprehensive account for for symmetric spectra is in
and for orthogonal spectra in
An account in terms of (∞,1)-category theory is in section 7.1 of
Discussion of simplicial ring spectra is in
See also the references at stable homotopy theory.