Geodesic
On wsq-primary ideals
Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 415-429
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We introduce weakly strongly quasi-primary (briefly, wsq-primary) ideals in commutative rings. Let $R$ be a commutative ring with a nonzero identity and $Q$ a proper ideal of $R$. The proper ideal $Q$ is said to be a weakly strongly quasi-primary ideal if whenever $0\neq ab\in Q$ for some $a,b\in R$, then $a^{2}\in Q$ or $b\in \sqrt {Q}.$ Many examples and properties of wsq-primary ideals are given. Also, we characterize nonlocal Noetherian von Neumann regular rings, fields, nonlocal rings over which every proper ideal is wsq-primary, and zero dimensional rings over which every proper ideal is wsq-primary. Finally, we study finite union of wsq-primary ideals.
We introduce weakly strongly quasi-primary (briefly, wsq-primary) ideals in commutative rings. Let $R$ be a commutative ring with a nonzero identity and $Q$ a proper ideal of $R$. The proper ideal $Q$ is said to be a weakly strongly quasi-primary ideal if whenever $0\neq ab\in Q$ for some $a,b\in R$, then $a^{2}\in Q$ or $b\in \sqrt {Q}.$ Many examples and properties of wsq-primary ideals are given. Also, we characterize nonlocal Noetherian von Neumann regular rings, fields, nonlocal rings over which every proper ideal is wsq-primary, and zero dimensional rings over which every proper ideal is wsq-primary. Finally, we study finite union of wsq-primary ideals.
DOI : 10.21136/CMJ.2023.0259-21
Classification : 05C25, 13A15, 13A99, 13F30
Keywords: primary ideal; weakly primary ideal; quasi-primary ideal; weakly 2-prime ideal; strongly quasi-primary ideal 
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     title = {On wsq-primary ideals},
     journal = {Czechoslovak Mathematical Journal},
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Aslankarayiğit Uğurlu, Emel; Bouba, El Mehdi; Tekir, Ünsal; Koç, Suat. On wsq-primary ideals. Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 415-429. doi: 10.21136/CMJ.2023.0259-21

[1] Almahdi, F. A. A., Tamekkante, M., Mamouni, A.: Rings over which every semi-primary ideal is 1-absorbing primary. Commun. Algebra 48 (2020), 3838-3845. | DOI | MR | JFM

[2] Anderson, D. F., (eds.), D. E. Dobbs: Zero-Dimensional Commutative Rings. Lecture Notes in Pure and Applied Mathematics. 171. Marcel Dekker, New York (1995). | MR | JFM

[3] Anderson, D. D., Smith, E.: Weakly prime ideals. Houston J. Math. 29 (2003), 831-840. | MR | JFM

[4] Anderson, D. D., Winders, M.: Idealization of a module. J. Commut. Algebra 1 (2009), 3-56. | DOI | MR | JFM

[5] Atiyah, M. F., Macdonald, I. G.: Introduction to Commutative Algebra. Addison-Wesley, Reading (1969). | MR | JFM

[6] Badawi, A.: On 2-absorbing ideals of commutative rings. Bull. Aust. Math. Soc. 75 (2007), 417-429. | DOI | MR | JFM

[7] Badawi, A., Sonmez, D., Yesilot, G.: On weakly $\delta$-semiprimary ideals of commutative rings. Algebra Colloq. 25 (2018), 387-398. | DOI | MR | JFM

[8] Badawi, A., Tekir, U., Yetkin, E.: On weakly 2-absorbing primary ideals of commutative rings. J. Korean Math. Soc. 52 (2015), 97-111. | DOI | MR | JFM

[9] Beddani, C., Messirdi, W.: 2-prime ideals and their applications. J. Algebra Appl. 15 (2016), Article ID 1650051, 11 pages. | DOI | MR | JFM

[10] Călugăreanu, G.: $UN$-rings. J. Algebra Appl. 15 (2016), Article ID 1650182, 9 pages. | DOI | MR | JFM

[11] Chen, J.: Infinite prime avoidance. Beitr. Algebra Geom. 62 (2021), 587-593. | DOI | MR | JFM

[12] Atani, S. Ebrahimi, Farzalipour, F.: On weakly primary ideals. Georgian Math. J. 12 (2005), 423-429. | DOI | MR | JFM

[13] Huckaba, J. A.: Commutative Rings with Zero Divisors. Monographs and Textbooks in Pure and Applied Mathematics 117. Marcel Dekker, New York (1988). | MR | JFM

[14] Koç, S.: On weakly 2-prime ideals in commutative rings. Commun. Algebra 49 (2021), 3387-3397. | DOI | MR | JFM

[15] Koc, S., Tekir, U., Ulucak, G.: On strongly quasi primary ideals. Bull. Korean Math. Soc. 56 (2019), 729-743. | DOI | MR | JFM

[16] Larsen, M. D., McCarthy, P. J.: Multiplicative Theory of Ideals. Pure and Applied Mathematics 43. Academic Press, New York (1971). | MR | JFM

[17] Lee, T.-K., Zhou, Y.: Reduced modules. Rings, Modules, Algebras and Abelian Groups Lecture Notes in Pure and Applied Mathematics 236. Marcel Dekker, New York (2004), 365-377. | MR | JFM

[18] Pakala, J. V., Shores, T. S.: On compactly packed rings. Pac. J. Math. 97 (1981), 197-201. | DOI | MR | JFM

[19] Redmond, S. P.: An ideal-based zero-divisor graph of a commutative ring. Commun. Algebra 31 (2003), 4425-4443. | DOI | MR | JFM

[20] Satyanarayana, M.: Rings with primary ideals as maximal ideals. Math. Scand. 20 (1967), 52-54. | DOI | MR | JFM

[21] Neumann, J. von: On regular rings. Proc. Natl. Acad. Sci. USA 22 (1936), 707-713. | DOI | JFM

[22] Wang, F., Kim, H.: Foundations of Commutative Rings and Their Modules. Algebra and Applications 22. Springer, Singapore (2016). | DOI | MR | JFM

[23] z, E. Yıldı, Ersoy, B. A., Tekir, Ü., Koç, S.: On $S$-Zariski topology. Commun. Algebra 49 (2021), 1212-1224. | DOI | MR | JFM

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