Nil$_{\ast}$-Artinian rings

Year 2023, Volume: 34 Issue: 34, 152 - 158, 10.07.2023

Xiaolei Zhang Wei Qı

https://doi.org/10.24330/ieja.1260486

Abstract

In this paper, we say a ring $R$ is Nil$_{\ast}$-Artinian if any
descending chain of nil ideals stabilizes. We first study
Nil$_{\ast}$-Artinian properties in terms of quotients,
localizations, polynomial extensions and idealizations, and then
study the transfer of Nil$_{\ast}$-Artinian rings to amalgamated
algebras. Besides, some examples are given to distinguish
Nil$_{\ast}$-Artinian rings, Nil$_{\ast}$-Noetherian rings and
Nil$_{\ast}$-coherent rings.

Keywords

Nil$_{\ast}$-Artinian ring, idealization, amalgamated algebra

References

• K. Adarbeh and S. Kabbaj, Trivial extensions subject to semi-regularity and semi-coherence, Quaest. Math., 43(1) (2020), 45-54.
• D. D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra, 1(1) (2009), 3-56.
• M. D'Anna, C. Finocchiaro and M. Fontana, Amalgamated algebras along an ideal, Commutative Algebra and its Applications, Walter de Gruyter, Berlin, (2009), 155-172.
• M. D'Anna and M. Fontana, An amalgamated duplication of a ring along an ideal: the basic properties, J. Algebra Appl., 6(3) (2007), 443-459.
• G. Donadze and V. Z. Thomas, Bazzoni-Glaz conjecture, J. Algebra, 420 (2014), 141-160.
• S. Glaz, Commutative Coherent Rings, Lecture Notes in Mathematics, 1371, Berlin, Spring-Verlag, 1989.
• J. A. Huckaba, Commutative Rings with Zero Divisors, Monographs and Textbooks in Pure and Applied Mathematics, 117, Marcel Dekker, Inc., New York, 1988.
• K. A. Ismaili, D. E. Dobbs and N. Mahdou, Commutative rings and modules that are Nil$_{\ast}$-coherent or special Nil$_{\ast}$-coherent, J. Algebra Appl., 16(10) (2017), 1750187 (24 pp).
• F. G. Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Algebra and Applications, 22, Singapore, Springer, 2016.
• Y. Xiang and L. Ouyang, Nil$_{\ast}$-coherent rings, Bull. Korean Math. Soc., 51(2) (2014), 579-594.
• X. L. Zhang, Nil$_{\ast}$-Noetherian rings, https://arxiv.org/abs/2205.11724.