An **associative ring spectrum**, or A-infinity ring, is an monoid object in the stable (infinity,1)-category of spectra with its smash product of spectra monoidal structure.

Equivalently, this should be the same as an ordinary monoid with respect to the symmetric monoidal smash product of spectra.

- A commutative associative ring spectrum is a commutative ring spectrum, or E-infinity ring.

the $(\infty,1)$-categorical description of associative ring spectra is in section 4.3 of

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