symmetric monoidal (∞,1)-category of spectra
In older literature, the term (commutative) ring spectrum refers to a (commutative) monoid in the stable homotopy category, hence to a spectrum equipped with a product operation, which is associative (and commutative) up to unspecified homotopy.
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.
In terms of modern homotopy theory the monoid structure alone is in general not quite appropriate, since one really needs A-∞ ring struture or even E-∞ ring structure. On the other hand, with a suitably symmetric monoidal smash product of spectra on the given category of spectra, this does follow.
A monoid-up-to-homotopy in the category of spectra for which the homotopies are coherent is called an -ring spectrum or just an -ring. These may be modeled as monoids with respect to the symmetric monoidal smash product of spectra.
Not every ring spectrum may be refined to an -ring spectrum.
An account in terms of (∞,1)-category theory is in section 7.1 of
An account in terms of model categories is in
See also the references at stable homotopy theory.