Czechoslovak Mathematical Journal, Vol. 72, No. 4, pp. 1133-1144, 2022

On quasi $n$-ideals of commutative rings

Adam Anebri, Najib Mahdou, Emel Aslankarayiğit Uğurlu

Received September 29, 2021.   Published online July 28, 2022.

Abstract:  Let $R$ be a commutative ring with a nonzero identity. In this study, we present a new class of ideals lying properly between the class of $n$-ideals and the class of $(2,n)$-ideals. A proper ideal $I$ of $R$ is said to be a quasi $n$-ideal if $\sqrt{I}$ is an $n$-ideal of $R.$ Many examples and results are given to disclose the relations between this new concept and others that already exist, namely, the $n$-ideals, the quasi primary ideals, the $(2,n)$-ideals and the $pr$-ideals. Moreover, we use the quasi $n$-ideals to characterize some kind of rings. Finally, we investigate quasi $n$-ideals under various contexts of constructions such as direct product, power series, idealization, and amalgamation of a ring along an ideal.
Keywords:  $n$-ideal; quasi $n$-ideal; $(2,n)$-ideal
Classification MSC:  13A15, 13A18

[1] D. D. Anderson, M. Winders: Idealization of a module. J. Commut. Algebra 1 (2009), 3-56. DOI 10.1216/JCA-2009-1-1-3 | MR 2462381 | Zbl 1194.13002
[2] A. Badawi, U. Tekir, E. Yetkin: On 2-absorbing primary ideals in commutative rings. Bull. Korean Math. Soc. 51 (2014), 1163-1173. DOI 10.4134/BKMS.2014.51.4.1163 | MR 3248714 | Zbl 1308.13001
[3] G. Călugăreanu: $UN$-rings. J. Algebra Appl. 15 (2016), Article ID 1650182, 9 pages. DOI 10.1142/S0219498816501826 | MR 3575972 | Zbl 1397.16037
[4] M. D'Anna, C. A. Finocchiaro, M. Fontana: Amalgamated algebras along an ideal. Commutative Algebra and Its Applications. Walter de Gruyter, Berlin (2009), 155-172. DOI 10.1515/9783110213188.155 | MR 2606283 | Zbl 1177.13043
[5] M. D'Anna, C. A. Finocchiaro, M. Fontana: Properties of chains of prime ideals in amalgamated algebras along an ideal. J. Pure Appl. Algebra 214 (2010), 1633-1641. DOI 10.1016/j.jpaa.2009.12.008 | MR 2593689 | Zbl 1191.13006
[6] M. D'Anna, M. Fontana: An amalgamated duplication of a ring along an ideal: The basic properties. J. Algebra Appl. 6 (2007), 443-459. DOI 10.1142/S0219498807002326 | MR 2337762 | Zbl 1126.13002
[7] L. Fuchs: On quasi-primary ideals. Acta Sci. Math. 11 (1947), 174-183. MR 0021541 | Zbl 0030.01101
[8] S. Hizem, A. Benhissi: Nonnil-Noetherian rings and the SFT property. Rocky Mt. J. Math. 41 (2011), 1483-1500. DOI 10.1216/RMJ-2011-41-5-1483 | MR 2838074 | Zbl 1242.13027
[9] R. Mohamadian: $r$-ideals in commutative rings. Turk. J. Math. 39 (2015), 733-749. DOI 10.3906/mat-1503-35 | MR 3395802 | Zbl 1348.13003
[10] M. Tamekkante, E. M. Bouba: $(2,n)$-ideals of commutative rings. J. Algebra Appl. 18 (2019), Article ID 1950103, 12 pages. DOI 10.1142/S0219498819501032 | MR 3954657 | Zbl 1412.13005
[11] U. Tekir, S. Koc, K. H. Oral: $n$-ideals of commutative rings. Filomat 31 (2017), 2933-2941. DOI 10.2298/FIL1710933T | MR 3639382 | Zbl 07418085
[12] U. Tekir, S. Koç, K. H. Oral, K. P. Shum: On 2-absorbing quasi-primary ideals in commutative rings. Commun. Math. Stat. 4 (2016), 55-62. DOI 10.1007/s40304-015-0075-9 | MR 3475842 | Zbl 1338.13007

Affiliations:   Adam Anebri, Najib Mahdou, Laboratory of Modelling and Mathematical Structures, Department of Mathematics, Faculty of Science and Technology of Fez, Box 2202, University S.M. Ben Abdellah Fez, Morocco, e-mail:,, Emel Aslankarayiğit Uğurlu (corresponding author), Department of Mathematics, Marmara University, Istanbul, Turkey, e-mail:

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