# Article

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Keywords:
$(\delta ,2)$-primary ideal; $2$-prime ideal; $\delta$-primary ideal
Summary:
Let $R$ be a commutative ring with nonzero identity, let $\mathcal {I(R)}$ be the set of all ideals of $R$ and $\delta \colon \mathcal {I(R)}\rightarrow \mathcal {I(R)}$ an expansion of ideals of $R$ defined by $I\mapsto \delta (I)$. We introduce the concept of $(\delta ,2)$-primary ideals in commutative rings. A proper ideal $I$ of $R$ is called a $(\delta ,2)$-primary ideal if whenever $a,b\in R$ and $ab\in I$, then $a^{2}\in I$ or $b^{2}\in \delta (I)$. Our purpose is to extend the concept of $2$-ideals to $(\delta ,2)$-primary ideals of commutative rings. Then we investigate the basic properties of $(\delta ,2)$-primary ideals and also discuss the relations among $(\delta ,2)$-primary, $\delta$-primary and $2$-prime ideals.
References:
[1] Anderson, D. D., Knopp, K. R., Lewin, R. L.: Ideals generated by powers of elements. Bull. Aust. Math. Soc. 49 (1994), 373-376. DOI 10.1017/S0004972700016488 | MR 1274517 | Zbl 0820.13004
[2] Anderson, D. D., Winders, M.: Idealization of a module. J. Commut. Algebra 1 (2009), 3-56. DOI 10.1216/JCA-2009-1-1-3 | MR 2462381 | Zbl 1194.13002
[3] Atiyah, M. F., Macdonald, I. G.: Introduction to Commutative Algebra. Addison-Wesley Publishing, Reading (1969). DOI 10.1201/9780429493621 | MR 0242802 | Zbl 0175.03601
[4] Badawi, A., Fahid, B.: On weakly 2-absorbing $\delta$-primary ideals of commutative rings. (to appear) in Georgian Math. J. DOI 10.1515/gmj-2018-0070 | MR 4168712
[5] Badawi, A., Sonmez, D., Yesilot, G.: On weakly $\delta$-semiprimary ideals of commutative rings. Algebra Colloq. 25 (2018), 387-398. DOI 10.1142/S1005386718000287 | MR 3843092 | Zbl 1401.13007
[6] Beddani, C., Messirdi, W.: 2-prime ideals and their applications. J. Algebra Appl. 15 (2016), Article ID 1650051, 11 pages. DOI 10.1142/S0219498816500511 | MR 3454713 | Zbl 1338.13038
[7] Gilmer, R.: Multiplicative Ideal Theory. Queen's Papers in Pure and Applied Mathematics 90, Queen's University, Kingston (1992). MR 1204267 | Zbl 0804.13001
[8] Groenewald, N. J.: A characterization of semi-prime ideals in near-rings. J. Aust. Math. Soc., Ser. A 35 (1983), 194-196. DOI 10.1017/S1446788700025660 | MR 0704424 | Zbl 0521.16030
[9] Huckaba, J. A.: Commutative Rings with Zero Divisors. Monographs and Textbooks in Pure and Applied Mathematics 117, Marcel Dekker, New York (1988). MR 0938741 | Zbl 0637.13001
[10] Kaplansky, I.: Commutative Rings. University of Chicago Press, Chicago (1974). MR 0345945 | Zbl 0296.13001
[11] Koc, S., Tekir, U., Ulucak, G.: On strongly quasi primary ideals. Bull. Korean Math. Soc. 56 (2019), 729-743. DOI 10.4134/BKMS.b180522 | MR 3960633 | Zbl 1419.13040
[12] Zhao, D.: $\delta$-primary ideals of commutative rings. Kyungpook Math. J. 41 (2001), 17-22. MR 1847432 | Zbl 1028.13001

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