In mathematics, a highly structured ring spectrum or ${\displaystyle A_{\infty }}$-ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology theory. A commutative version of an ${\displaystyle A_{\infty }}$-ring is called an ${\displaystyle E_{\infty }}$-ring. While originally motivated by questions of geometric topology and bundle theory, they are today most often used in stable homotopy theory.

Highly structured ring spectra have better formal properties than multiplicative cohomology theories – a point utilized, for example, in the construction of topological modular forms, and which has allowed also new constructions of more classical objects such as Morava K-theory. Beside their formal properties, ${\displaystyle E_{\infty }}$-structures are also important in calculations, since they allow for operations in the underlying cohomology theory, analogous to (and generalizing) the well-known Steenrod operations in ordinary cohomology. As not every cohomology theory allows such operations, not every multiplicative structure may be refined to an ${\displaystyle E_{\infty }}$-structure and even in cases where this is possible, it may be a formidable task to prove that.

The rough idea of highly structured ring spectra is the following: If multiplication in a cohomology theory (analogous to the multiplication in singular cohomology, inducing the cup product) fulfills associativity (and commutativity) only up to homotopy, this is too lax for many constructions (e.g. for limits and colimits in the sense of category theory). On the other hand, requiring strict associativity (or commutativity) in a naive way is too restrictive for many of the wanted examples. A basic idea is that the relations need only hold up to homotopy, but these homotopies should fulfill again some homotopy relations, whose homotopies again fulfill some further homotopy conditions; and so on. The classical approach organizes this structure via operads, while the recent approach of Jacob Lurie deals with it using ${\displaystyle \infty }$-operads in ${\displaystyle \infty }$-categories. The most widely used approaches today employ the language of model categories.^{[citation needed]}

All these approaches depend on building carefully an underlying category of spectra.

The theory of operads is motivated by the study of loop spaces. A loop space ΩX has a multiplication

by composition of loops. Here the two loops are sped up by a factor of 2 and the first takes the interval [0,1/2] and the second [1/2,1]. This product is not associative since the scalings are not compatible, but it is associative up to homotopy and the homotopies are coherent up to higher homotopies and so on. This situation can be made precise by saying that ΩX is an algebra over the little interval operad. This is an example of an ${\displaystyle A_{\infty }}$-operad, i.e. an operad of topological spaces which is homotopy equivalent to the associative operad but which has appropriate "freeness" to allow things only to hold up to homotopy (succinctly: any cofibrant replacement of the associative operad). An ${\displaystyle A_{\infty }}$-ring spectrum can now be imagined as an algebra over an ${\displaystyle A_{\infty }}$-operad in a suitable category of spectra and suitable compatibility conditions (see May, 1977).

For the definition of ${\displaystyle E_{\infty }}$-ring spectra essentially the same approach works, where one replaces the ${\displaystyle A_{\infty }}$-operad by an ${\displaystyle E_{\infty }}$-operad, i.e. an operad of contractible topological spaces with analogous "freeness" conditions. An example of such an operad can be again motivated by the study of loop spaces. The product of the double loop space ${\displaystyle \Omega ^{2}X}$ is already commutative up to homotopy, but this homotopy fulfills no higher conditions. To get full coherence of higher homotopies one must assume that the space is (equivalent to) an n-fold loopspace for all n. This leads to the in ${\displaystyle \infty }$-cube operad of infinite-dimensional cubes in infinite-dimensional space, which is an example of an ${\displaystyle E_{\infty }}$-operad.

The above approach was pioneered by J. Peter May. Together with Elmendorf, Kriz and Mandell he developed in the 90s a variant of his older definition of spectra, so called S-modules (see Elmendorf et al., 2007). S-modules possess a model structure, whose homotopy category is the stable homotopy category. In S-modules the category of modules over an ${\displaystyle A_{\infty }}$-operad and the category of monoids are Quillen equivalent and likewise the category of modules over an ${\displaystyle E_{\infty }}$-operad and the category of commutative monoids. Therefore, is it possible to define ${\displaystyle A_{\infty }}$-ring spectra and ${\displaystyle E_{\infty }}$-ring spectra as (commutative) monoids in the category of S-modules, so called (commutative) S-algebras. Since (commutative) monoids are easier to deal with than algebras over complicated operads, this new approach is for many purposes more convenient. It should, however, be noted that the actual construction of the category of S-modules is technically quite complicated.