On the cyclic homology of certain universal differential graded algebras

Let $p$ be an odd prime and $R$ a $p$-torsion-free commutative ${\mathbb{Z}}_{(p)}$-algebra. We compute the periodic cyclic homology over $R$ of the universal differential graded algebra $R\mathrm{/}\mathrm{/}p$ which is obtained from $R$ by universally killing $p$. We furthermore compute the cyclic and negative cyclic homologies of $R\mathrm{/}\mathrm{/}p$ over $R$ in infinitely many degrees.