Volume 34, Issue 1
Extensions of Modules with ACC on $d$-Annihilators

Commun. Math. Res., 34 (2018), pp. 23-35.

Published online: 2019-12

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• Abstract

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.

• Keywords

triangular matrix extension, Ore extension, acc on $d$-annihilator

16S99

ouyanglqtxy@163.com (Lunqun Ouyang)

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@Article{CMR-34-23, author = {Ouyang , Lunqun and Zhou , Qiong and Liu , Jinwang and Xiang , Yueming}, title = {Extensions of Modules with ACC on $d$-Annihilators}, journal = {Communications in Mathematical Research }, year = {2019}, volume = {34}, number = {1}, pages = {23--35}, abstract = {

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.

}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2018.01.03}, url = {http://global-sci.org/intro/article_detail/cmr/13489.html} }
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 -

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.

Lunqun Ouyang, Qiong Zhou, Jinwang Liu & Yueming Xiang. (2019). Extensions of Modules with ACC on $d$-Annihilators. Communications in Mathematical Research . 34 (1). 23-35. doi:10.13447/j.1674-5647.2018.01.03
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