Browse > Article
http://dx.doi.org/10.4134/BKMS.b200614

ON PSEUDO 2-PRIME IDEALS AND ALMOST VALUATION DOMAINS  

Koc, Suat (Department of Mathematics Marmara University)
Publication Information
Bulletin of the Korean Mathematical Society / v.58, no.4, 2021 , pp. 897-908 More about this Journal
Abstract
In this paper, we introduce the notion of pseudo 2-prime ideals in commutative rings. Let R be a commutative ring with a nonzero identity. A proper ideal P of R is said to be a pseudo 2-prime ideal if whenever xy ∈ P for some x, y ∈ R, then x2n ∈ Pn or y2n ∈ Pn for some n ∈ ℕ. Various examples and properties of pseudo 2-prime ideals are given. We also characterize pseudo 2-prime ideals of PID's and von Neumann regular rings. Finally, we use pseudo 2-prime ideals to characterize almost valuation domains (AV-domains).
Keywords
Prime ideal; 2-prime ideal; pseudo 2-prime ideal; valuation domain; almost valuation domain;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 D. D. Anderson and M. Zafrullah, Almost B'ezout domains, J. Algebra 142 (1991), no. 2, 285-309. https://doi.org/10.1016/0021-8693(91)90309-V   DOI
2 A. Badawi, On divided commutative rings, Comm. Algebra 27 (1999), no. 3, 1465-1474. https://doi.org/10.1080/00927879908826507   DOI
3 C. Beddani and W. Messirdi, 2-prime ideals and their applications, J. Algebra Appl. 15 (2016), no. 3, 1650051, 11 pp. https://doi.org/10.1142/S0219498816500511   DOI
4 D. E. Dobbs, Divided rings and going-down, Pacific J. Math. 67 (1976), no. 2, 353-363. http://projecteuclid.org/euclid.pjm/1102817497   DOI
5 J. A. Huckaba, Commutative rings with zero divisors, Monographs and Textbooks in Pure and Applied Mathematics, 117, Marcel Dekker, Inc., New York, 1988.
6 C. Jayaram and Tekir, von Neumann regular modules, Comm. Algebra 46 (2018), no. 5, 2205-2217. https://doi.org/10.1080/00927872.2017.1372460   DOI
7 S. Koc, U. Tekir, and G. Ulucak, On strongly quasi primary ideals, Bull. Korean Math. Soc. 56 (2019), no. 3, 729-743. https://doi.org/10.4134/BKMS.b180522   DOI
8 N. Mahdou, A. Mimouni, and M. A. S. Moutui, On almost valuation and almost Bezout rings, Comm. Algebra 43 (2015), no. 1, 297-308. https://doi.org/10.1080/00927872.2014.897586   DOI
9 U. Tekir, G. Ulucak, and S. Koc, On divided modules, Iran. J. Sci. Technol. Trans. A Sci. 44 (2020), no. 1, 265-272. https://doi.org/10.1007/s40995-020-00827-1   DOI
10 J. von Neumann, On regular rings, Proceedings of the National Academy of Sci. 22 (1936), no. 12, 707-713.   DOI
11 A. Badawi, U. Tekir, and E. Yetkin, On 2-absorbing primary ideals in commutative rings, Bull. Korean Math. Soc. 51 (2014), no. 4, 1163-1173. https://doi.org/10.4134/BKMS.2014.51.4.1163   DOI
12 S. C eken Gezen, On M-coidempotent elements and fully coidempotent modules, Comm. Algebra 48 (2020), no. 11, 4638-4646. https://doi.org/10.1080/00927872.2020.1768266   DOI
13 R. Jahani-Nezhad and F. Khoshayand, Almost valuation rings, Bull. Iranian Math. Soc. 43 (2017), no. 3, 807-816.
14 R. Nikandish, M. J. Nikmehr, and A. Yassine, More on the 2-prime ideals of commutative rings, Bull. Korean Math. Soc. 57 (2020), no. 1, 17-126.
15 M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, MA, 1969.
16 A. Ayache and D. E. Dobbs, Strongly divided pairs of integral domains, in Advances in commutative algebra, 63-92, Trends Math, Birkhauser/Springer, Singapore, 2019. https://doi.org/10.1007/978-981-13-7028-1_4
17 A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc. 75 (2007), no. 3, 417-429. https://doi.org/10.1017/S0004972700039344   DOI
18 D. D. Anderson, S. Chun, and J. R. Juett, Module-theoretic generalization of commutative von Neumann regular rings, Comm. Algebra 47 (2019), no. 11, 4713-4728. https://doi.org/10.1080/00927872.2019.1593427   DOI
19 D. D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra 1 (2009), no. 1, 3-56. https://doi.org/10.1216/JCA-2009-1-1-3   DOI
20 A. Badawi and D. E. Dobbs, On locally divided rings and going-down rings, Comm. Algebra 29 (2001), no. 7, 2805-2825. https://doi.org/10.1081/AGB-100104988   DOI
21 D. E. Dobbs, A. El Khalfi, and N. Mahdou, Trivial extensions satisfying certain valuation-like properties, Comm. Algebra 47 (2019), no. 5, 2060-2077. https://doi.org/10.1080/00927872.2018.1527926   DOI
22 L. Fuchs, On quasi-primary ideals, Acta Univ. Szeged. Sect. Sci. Math. 11 (1947), 174-183.
23 Max. D. Larsen and P. J. McCarthy, Multiplicative Theory of Ideals, Academic Press, New York, 1971.