nLab
commutative ring spectrum

Idea

An ordinary (commutative) ring is a (commutative) monoid or algebra object internal to the symmetric monoidal category Ab of abelian groups.

The role that Ab plays among ordinary categories is played by the stable (infinity,1)-category of spectra $\mathrm{Sp}$ among (infinity,1)-categories.

A commutative ring spectrum, or E-infinity ring, is accordingly a commutative algebra in an (infinity,1)-category.

Definition

A commutative ring spectrum is a commutative algebra object in the symmetric monoidal (infinity,1)-category of spectra.

References

reference to standard literature on $E_{\mathrm{\infty }}$ things goes here …

then..

the $(\mathrm{\infty },1)$-categorical perspective on commutative ring spectra is the topic of

section 7 of

• Jacob Lurie, Commutative algebra.