Abstract
In this paper, we prove that a finitely embedded R-module M is Artinian if and only if for every prime ideal \(\mathfrak {p}\) of R with \((0:_RM)\subseteq \mathfrak {p}\), there exists a submodule \(N^\mathfrak {p}\) of M such that \(M/N^\mathfrak {p}\) is finitely embedded and \(M[\mathfrak {p}]\subseteq N^\mathfrak {p}\subseteq (0:_M\mathfrak {p})\), where \(M[\mathfrak {p}]=\bigcap \nolimits _{s\in R {\setminus } \mathfrak {p}}s(0:_M\mathfrak {p}).\)
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Acknowledgements
The first author was supported by the National Natural Science Foundation of China (No. 12061001). The second author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2021R1I1A3047469). The third author was supported by National Natural Science Foundation of China (No. 12201361).
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Zhang, X., Kim, H. & Qi, W. A note on a Cohen-type theorem for Artinian modules. Beitr Algebra Geom 64, 1107–1110 (2023). https://doi.org/10.1007/s13366-022-00671-x
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DOI: https://doi.org/10.1007/s13366-022-00671-x