TITLE: solid commutative ring spectra
QUESTION [5 upvotes]: Let $R$ be a discrete (i.e. an ordinary) commutative ring and let $HR\rightarrow T$ be a map of $E_{\infty}$-ring spectra where $HR$ is the associated Eilenberg-Mac Lane ring spectrum. We say that $T$ is $R$-solid (in the derived sense) if the induced map of $E_{\infty}$-ring spectra
$$T\wedge_{HR}^{\mathbb{L}}T\rightarrow T$$ is a weak equivalence. 
Are there some nontrivial examples of such solid ring spectra? By "nontrivial" I mean that $T$ is not discrete. The discrete case (over $\mathbf{Z}$) is classified here .

REPLY [4 votes]: Here is an example of such pair $T$ and $R$. 

$k$ is a finite field.
$G$ is the Lie group $SO(2)$ and $G^{\delta}$ the same group but with dicrete topology.
$R$ is the group algebra $k[G^{\delta}]$.
$T$ is $C_{\ast}(G,k)$ the singular chain complex associated to $G$.

Then the natural map $k[G^{\delta}]\rightarrow C_{\ast}(G,k)$ of $E_{\infty}$-algebras induces a quasi-isomorphism 
$$ C_{\ast}(G,k)\otimes_{k[G^{\delta}]}^{\mathbf{L}} C_{\ast}(G,k)\rightarrow C_{\ast}(G,k)$$ of $E_{\infty}$-algebras.