TY - JOUR
T1 - Extensions of Modules with ACC on $d$-Annihilators
AU - Ouyang , Lunqun
AU - Zhou , Qiong
AU - Liu , Jinwang
AU - Xiang , Yueming
JO - Communications in Mathematical Research 
VL - 1
SP - 23
EP - 35
PY - 2019
DA - 2019/12
SN - 34
DO - http://doi.org/10.13447/j.1674-5647.2018.01.03
UR - https://global-sci.org/intro/article_detail/cmr/13489.html
KW - triangular matrix extension, Ore extension, acc on $d$-annihilator
AB - <p style="text-align: justify;">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&nbsp; modules with acc on $d$-annihilators are extended to a more general setting.</p>