20025

Nil$_{\ast}$-Artinian rings

ISSN: 1306-6048
Yayın Aralığı: Yılda 2 Sayı
Başlangıç: 2007
Yayıncı: Abdullah HARMANCI

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,

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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.