A unitary right $R$-module $M_R$ satisfies acc on $d$-annihilators if for every sequence $(a_n)_n$ of elements of $R$ the ascending chain ${\rm Ann}_M(a_1)\subseteq{\rm Ann}_M(a_1a_2)\subseteq{\rm Ann}_M(a_1a_2a_3)\subseteq\cdots$ of submodules of $M_R$ stabilizes. In this paper we first investigate some triangular matrix extensions of modules with acc on $d$-annihilators. Then we show that under some additional conditions, the Ore extension module $M[x]_{R[x;\alpha,\delta]}$ over the Ore extension ring $R[x;\,\alpha,\delta]$ satisfies acc on $d$-annihilators if and only if the module $M_R$ satisfies acc on $d$-annihilators. Consequently, several known results regarding modules with acc on $d$-annihilators are extended to a more general setting.
