For $R$ any ring and $H R$ its Eilenberg-MacLane spectrum -- a ring spectrum -- there is an equivalence between the $\infty$-categories of $H R$-module spectra and that of unbounded chain complexes of $R$-modules - a stable version of the Dold-Kan correspondence.

At least in good cases such as $R = \mathbb{Z}$ this refines also to an equivalence between $H \mathbb{Z}$-algebra spectra and unbounded dg-rings -- a stable version of the monoidal Dold-Kan correspondence. (A presentation of this by a Quillen equivalence has been given by Shipley, see http://ncatlab.org/nlab/show/algebra+spectrum).

Now for $X$ a sufficiently nice topological space, one would hope that under this equivalence the "spectrum of integral homology chains" $(\Sigma^\infty_+ \Omega X) \wedge H \mathbb{Z}$ on the loop space of $X$ is identified, up to equivalence, with the ordinary chain complex of singular homology chains $C_\bullet(\Omega X, \mathbb{Z})$, both as $\infty$-modules and as $\infty$-algebra objects.

This must be easy to see, but I am a little stuck. Can anyone help?