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Endo-Noetherian rings. (English) Zbl 1439.13045

A commutative ring \(R\) is said to be endo-Noetherian if the chain of annihilators \(\mathrm{ann}(a_{1})\subseteq\mathrm{ann}(a_{2})\dots\) stabilizes for any sequence \((a_{k})_{k}\) of elements of \(R\) [M. A. Ndiaye and C. T. Gueye, Int. J. Appl. Math. 86, No. 5, 871–881 (2013; doi:10.12732/ijpam.v86i5.10)]. In this paper under review, the authors investigate the properties of endo-Noetherian rings, specially the behaviour of endo-Noetherian rings under localization, in factor rings, in product of rings, in polynomial rings over Armendariz rings and in power series rings over power serieswise Armendariz rings.

MSC:

13E99 Chain conditions, finiteness conditions in commutative ring theory
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13F25 Formal power series rings
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References:

[1] Anderson, DD; Dumitrescu, T., \(S\)-Noetherian rings, Commun. Algebra, 30, 4407-4416 (2002) · Zbl 1060.13007 · doi:10.1081/AGB-120013328
[2] Anderson, DD; Winders, M., Idealization of a module, Rocky Mt. J. Math., 1, 3-56 (2009) · Zbl 1194.13002
[3] Benhissi, A., Chain condition on annihilators and strongly Hopfian property in Hurwitz series ring, Algebra Colloq., 21, 635-646 (2014) · Zbl 1304.13041 · doi:10.1142/S1005386714000583
[4] D’Anna, M.; Fontana, M., An amalgamated duplication of a ring along an ideal, the basic properties, J. Algebra Appl., 6, 443-459 (2007) · Zbl 1126.13002 · doi:10.1142/S0219498807002326
[5] D’Anna, M.; Fontana, M., An amalgamated duplication of a ring along a multiplicative canonical ideal, Arkiv Mat., 6, 241-252 (2007) · Zbl 1143.13002 · doi:10.1007/s11512-006-0038-1
[6] Finocchiaro, CA, Prüfer-like conditions on an amalgamated algebra along an ideal, Houst. J. Math., 40, 63-79 (2013) · Zbl 1297.13002
[7] Facchinia, A.; Nazemian, Z., Modules with chain conditions up to isomorphism, J. Algebra, 453, 578-601 (2016) · Zbl 1408.16017 · doi:10.1016/j.jalgebra.2016.01.025
[8] Gilmer, R.; Heinzer, W., The Laskerian property, power series rings and Noetherian spectra, Proc. Am. Math. Soc., 79, 13-16 (1980) · Zbl 0447.13009 · doi:10.1090/S0002-9939-1980-0560575-6
[9] Hamed, A., Gouaid, B., Benhissi, A.: Rings satisfying the strongly Hopfian and \(S\)-strongly Hopfian properties. Math. Reports (in press) · Zbl 1524.13066
[10] Hamed, A.; Hizem, S., \(S\)-Noetherian rings of the forms \(\cal{A}[X]\) and \(\cal{A}[[X]]\), Commun. Algebra, 43, 3848-3856 (2015) · Zbl 1329.13014 · doi:10.1080/00927872.2014.924127
[11] Heinzer, W.; Ohm, J., On the Noetherian-like rings of E. G. Evans, Proc. Am. Math. Soc., 34, 73-74 (1972) · Zbl 0239.13014 · doi:10.1090/S0002-9939-1972-0294316-2
[12] Hizem, S., Formal power series over strongly hopfian rings, Commun. Algebra, 39, 279-291 (2011) · Zbl 1208.13008 · doi:10.1080/00927870903527527
[13] Hmaimou, A.; Kaidi, A.; Campos, ES, Generalized fitting modules and rings, J. Algebra, 308, 199-214 (2007) · Zbl 1115.16002 · doi:10.1016/j.jalgebra.2006.07.010
[14] Kim, N.; Lee, Y., Armendariz rings and reduced rings, J. Algebra, 223, 477-488 (2000) · Zbl 0957.16018 · doi:10.1006/jabr.1999.8017
[15] Kim, NK; Lee, KH; Lee, Y., Power series rings satisfying a zero divisor property, Commun. Algebra, 34, 2205-2218 (2006) · Zbl 1121.16037 · doi:10.1080/00927870600549782
[16] Lu, CP, Modules and rings satisfying ACCR, Proc. Am. Math. Soc., 117, 5-10 (1993) · Zbl 0768.13002 · doi:10.1090/S0002-9939-1993-1104398-7
[17] Ndiaye, MA; Gueye, CT, On commutative EKFN-ring with ascending chain condition on annihilators, Int. J. Appl. Math., 86, 871-881 (2013)
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