A solid ring is a ring $R$ such that the multiplication
$R\otimes_{\mathbb{Z}} R \to R$ is an isomorphism.

These were classified by Bousfield and Kan; they are

subrings of $R\subseteq\mathbb{Q}$,

$\mathbb{Z}/n$,

products $R\times \mathbb{Z}/n$ with $R\subseteq \mathbb{Q}$ and every divisor of $n$ invertible in $R$

colimits of these.

I wonder how small the list gets if I put the additional constraint that $\mathrm{Tor}_{\mathbb{Z}}(R,R) = 0$.

REFERENCE: Bousfield, A. K.; Kan, D. M. The core of a ring. J. Pure Appl. Algebra 2 (1972), 73–81.