  We consider a class of self-injective special biserial algebras $\Lambda_N$
over a field $K$ and show that the Hochschild cohomology ring of $\Lambda_N$ is
a finitely generated $K$-algebra. Moreover the Hochschild cohomology ring of
$\Lambda_N$ modulo nilpotence is a finitely generated commutative $K$-algebra
of Krull dimension two. As a consequence the conjecture of Snashall-Solberg
\cite{SS}, concerning the Hochschild cohomology ring modulo nilpotence, holds
for this class of algebras.
