An $A_\infty$-ring is a monoid in an (∞,1)-category in an additive (that is, stable) (∞,1)-category. Alternatively one can take a model which is a (non-homotopic) additive monoidal category, but the monoid is replaced by an algebra over a resolution of the associative operad.

Another version of the $A_\infty$-ring is simply what is usually called the $A_\infty$-algebra in the case when the ground ring is the ring of integers. See

Gerald Dunn, Lax operad actions and coherence for monoidal $n$-Categories, $A_{\infty}$ rings and modules, Theory Appl. Cat. 1997, n.4 (TAC)

Last revised on May 27, 2016 at 13:39:09.
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