Hochschild cohomology of the quadratic monomial algebra Nₘ

Let Nₘ (R) = \ (a₈₉) { Mₘ (R) a₁₁ = a₂₂ = = a₌₌ and a₈₉ = 0 for any i > j \} for a commutative ring R. Then Nₘ (R) is a quadratic monomial algebra over R. We calculate HH^ (Nₘ (R), Mₘ (R) / Nₘ (R) ) as R-modules. We also determine the R-algebra structure of the Hochschild cohomology ring HH^ (Nₘ (R), Nₘ (R) ). For m 3, HH^ (Nₘ (R), Nₘ (R) ) is an infinitely generated algebra over R and has no Batalin-Vilkovisky algebra structure giving the Gerstenhaber bracket.