  Let ${\rm N}_m(R) = \{ (a_{ij}) \in {\rm M}_m(R) \mid a_{11} = a_{22} =
\cdots = a_{mm} \mbox{ and } a_{ij} = 0 \mbox{ for any } i > j \}$ for a
commutative ring $R$. Then ${\rm N}_m(R)$ is a quadratic monomial algebra over
$R$. We calculate ${\rm HH}^{\ast}({\rm N}_m(R), {\rm M}_m(R)/{\rm N}_m(R))$ as
$R$-modules. We also determine the $R$-algebra structure of the Hochschild
cohomology ring ${\rm HH}^{\ast}({\rm N}_m(R), {\rm N}_m(R))$. For $m \ge 3$,
${\rm HH}^{\ast}({\rm N}_m(R), {\rm N}_m(R))$ is an infinitely generated
algebra over $R$ and has no Batalin-Vilkovisky algebra structure giving the
Gerstenhaber bracket.
