  Let $B$ be a block algebra of a group algebra $FG$ of a finite group $G$ over
a field $F$ of characteristic $p>0$. This paper studies ring theoretic
properties of the representation ring $T^\Delta(B,B)$ of perfect
$p$-permutation $(B,B)$-bimodules and properties of the $k$-algebra
$k\otimes_\mathbb{Z} T^\Delta(B,B)$, for a field $k$. We show that if the
Cartan matrix of $B$ has $1$ as an elementary divisor then $[B]$ is not
primitive in $T^\Delta(B,B)$. If $B$ has cyclic defect groups we determine a
primitive decomposition of $[B]$ in $T^\Delta(B,B)$. Moreover, if $k$ is a
field of characteristic different from $p$ and $B$ has cyclic defect groups of
order $p^n$ we describe $k\otimes_\mathbb{Z} T^\Delta(B,B)$ explicitly as a
direct product of a matrix algebra and $n$ group algebras.
