Hochschild cohomology of the quadratic monomial algebra ${\mathrm{N}}_m$

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