Is there a good description of the homotopy type of a free simplicial ring (or simplicial $R$-algebra) on a given simplicial set, in terms of the homotopy type of that simplicial set?


(This is mostly an idle question, but also motivated by the fact that it is a theorem of Milnor that a similar construction with the free group gives a model for the $\Omega \Sigma X$, perhaps believable in view of the fact that $\Omega \Sigma$ is supposed to be the left adjoint from spaces into grouplike $A\_\infty $-(i.e., with a coherently associative multiplication law) spaces.)



What is the homotopy type of a free simplicial ring? 
Let $R[-]$ be the free $R$-module functor, from sets to $R$-modules, and $T\_R$ the free (tensor) $R$-algebra functor, from $R$-modules to $R$-algebras. The free $R$ algebra functor from sets to $R$-algebras is the compoisite $T\_RR[-]$.


Given a simplicial set $X$, the homotopy type of $R[X]$ is well understood. It is a product of Eilenberg-MacLane spaces, and its homotopy groups are the homology groups of $X$ with coefficients in $R$,


$$R[X]\simeq \prod\_{n\geq 0}K(H\_n(X,R),n).$$


Since 


$$T\_RM=\bigoplus\_{m\geq 0}M^{\otimes m}$$ 


we have 


$$T\_RR[X]=\bigoplus\_{m\geq 0}R[X\times\stackrel{m}\cdots\times X]$$ 


therefore


$$T\_RR[X]\simeq \prod\_{n\geq 0}K\left(\bigoplus\_{m\geq 0}H\_n(X\times\stackrel{m}\cdots\times X,R),n\right).$$


