A multiplicative comparison of Mac Lane homology and topological Hochschild homology

Let Q denote Mac Lane's Q-construction and circle times the smash product of spectra. We construct an equivalence Q(R) similar or equal to Z circle times R in the category of A(infinity) ring spectra for any ring R, thus proving a conjecture of Fiedorowicz, Pirashvili, Schwanzl, Vogt and Waldhausen. More precisely, we construct a symmetric monoidal structure on Q (in the infinity-categorical sense) extending the usual monoidal structure, for which we prove an equivalence Q (-) similar or equal to Z circle times- as symmetric monoidal functors. From this, we obtain a new proof of the equivalence HML(R,M) similar or equal to THH (R,M) originally proved by Pirashvili and Waldhausen. This equivalence is in fact made symmetric monoidal, and so it also provides a proof of the equivalence HML(R) similar or equal to THH (R) as E-infinity ring spectra, when R is a commutative ring.