Highly structured ring spectrum

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.