Derived ring theory

In algebra, the derived ring theory is a generalization of ring theory. Precisely, it studies (associative) ring spectra such as the complex K-theory spectrum. It generalizes the traditional ring theory to a situation where the underlying set of a ring is replaced by a spectrum in topology and where the ring of the integers is replaced by the sphere spectrum. It also provides the foundations for derived algebraic geometry, just as commutative rings are foundational for ordinary algebraic geometry.
Over characteristic zero, the commutative theory, also known as derived commutative algebra, is equivalent to the theory of commutative differential graded algebras. (cf. .)
A derived ring is sometimes also called a brave new ring.
Foundations
Let R be a commutative ring. The ∞-category of over R can be identified with that of differential graded algebras over R. If R contains Q, then the ∞-category of over R can be identified with that of commutative differential graded algebras over R.
Note that E1-spaces are ; see also ,
Let R be a commutative ring. The ∞-category of connective E1-algebras over R can be identified with that of simplicial algebras over R. If R contains Q, then the ∞-category of connective E∞-algebras over R can be identified with that of simplicial commutative algebras over R.
Basic concepts
Module spectrum
Cofibrant replacement
A cofibrant replacement is roughly like a resolution.

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