Absorbing ideals in commutative rings: a survey. (English) Zbl 1529.13004
Chabert, Jean-Luc (ed.) et al., Algebraic, number theoretic, and topological aspects of ring theory. Selected papers based on the cancelled conference on rings and polynomials, July 2020, and the fourth international meeting on integer-valued polynomials and related topics, CIRM, Luminy, France, July 19–24, 2021. Cham: Springer. 51-60 (2023).
Summary: Let \(R\) be a commutative ring with \(1 \neq 0\) and \(I\) be a proper ideal of \(R\). Then \(I\) is called a 2-absorbing ideal of \(R\) if whenever \(abc \in I\) for some \(a, b, c \in R\), then \(ab \in I\) or \(bc \in I\) or \(ac \in I\). This article surveys recent developments on 2-absorbing ideals and their generalizations in commutative rings.
For the entire collection see [Zbl 1515.13002].
For the entire collection see [Zbl 1515.13002].
MSC:
| 13A15 | Ideals and multiplicative ideal theory in commutative rings |
References:
| [1] | M.T. Ahmed, T. Dumitrescu and M. Azeem Khadam, Commutative rings with absorbing factorization , Comm. Algebra, 48(12), 5067-5075 (2020). · Zbl 1443.13002 · doi:10.1080/00927872.2020.1778714 |
| [2] | D.F. Anderson and A. Badawi , \( On n\)-absorbing ideals of commutative rings, Comm. Algebra, 39, 1646-1672 (2011). · Zbl 1232.13001 · doi:10.1080/00927871003738998 |
| [3] | D.F. Anderson and A. Badawi , Von Neumann regular and related elements in commutative rings, Algebra Colloq., 19(Spec 1), 1017-1040 (2012). · Zbl 1294.13029 |
| [4] | D.F Anderson and A. Badawi , \( On (m, n)\)-closed ideals of commutative rings , J. Algebra Appl.,16(1), 2016. |
| [5] | D.F. Anderson, A. Badawi and B. Fahid, \( Weakly (m, n)\)-closed ideals and \((m, n)\)-Von Neumann regular rings, J. Korean Math. Soc., 55(5), 1031-1043 (2018). · Zbl 1401.13008 |
| [6] | D.F. Anderson and D.E. Dobbs, Pairs of rings with the same prime ideals, Canad. J. Math., 32, 362-384 (1980). · Zbl 0406.13001 · doi:10.4153/CJM-1980-029-2 |
| [7] | D.D. Anderson and J.L. Mott, Cohen-Kaplansky domains: Integral domains with a finite number of irreducible elements, J. Algebra, 148(1), 17-41 (1992). · Zbl 0773.13005 · doi:10.1016/0021-8693(92)90234-D |
| [8] | D.D. Anderson and E. Smith, Weakly prime ideals, Houston J. Math., 29(4), 831-840 (2008). · Zbl 1086.13500 |
| [9] | A. Badawi, \(n\)-Absorbing Ideals of Commutative Rings and Recent Progress on Three Conjectures: A Survey. In Rings, Polynomials, and Modules, edited by Marco Fontana, Sophie Frisch, Sarah Glaz, Francesca Tartarone, and Paolo Zanardo, Springer, 33-52 (2017). · Zbl 1390.13005 |
| [10] | A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc. 75, 417-429 (2007). · Zbl 1120.13004 · doi:10.1017/S0004972700039344 |
| [11] | A. Badawi, On weakly semiprime ideals of commutative rings, Beitr Algebra Geom., 57, 589-597 (2016). · Zbl 1349.13005 · doi:10.1007/s13366-016-0283-9 |
| [12] | A. Badawi, On commutative divided rings, Comm. Algebra, 27(3), 1465-1474 (1999). · Zbl 0923.13001 · doi:10.1080/00927879908826507 |
| [13] | A. Badawi and E.Y. Celikel, On 1-absorbing primary ideals of commutative rings, J. Algebra Appl., 19(6), 2050111 (2020). · Zbl 1440.13008 |
| [14] | A. Badawi and A.Y. Darani , On weakly 2-absorbing ideals of commutative rings, Houston J. Math., 39(2), 441-452 (2013). · Zbl 1278.13001 |
| [15] | A. Badawi and D. E. Dobbs, On locally divided rings and going-down rings, Comm. Algebra, 29(7), 2805-2825 (2001). · Zbl 1104.13301 · doi:10.1081/AGB-4988 |
| [16] | A. Badawi , B. Fahid, On weakly 2-absorbing \(δ\)-primary ideals of commutative rings, Georgian. Math. J, 27(4), 503-516 (2020). · Zbl 1457.13002 · doi:10.1515/gmj-2018-0070 |
| [17] | A. Badawi, M. Issoual and N. Mahdou, \( On n\)-absorbing ideals and \((m, n)\)-closed ideals in trivial ring extensions of commutative rings, J. Alg. Appl., 18(7), 2019, 1950123, 19 pp (2019). · Zbl 1448.13006 |
| [18] | A. Badawi , U. Tekir and E.Yetkin, On 2-absorbing primary ideals in commutative rings, Bull. Korean. Math. Soc., 51(4), 1163-1173 (2014). · Zbl 1308.13001 · doi:10.4134/BKMS.2014.51.4.1163 |
| [19] | A. Badawi, U. Tekir, and E. Yetkin, On weakly 2-absorbing primary ideals of commutative rings, J. Korean Math. Soc., 52(1), 97-111 (2015). · Zbl 1315.13008 · doi:10.4134/JKMS.2015.52.1.097 |
| [20] | A.Badawi, D. Sonmez, G.Yesilot, On weakly \(δ\)-semiprimary ideals of commutative rings, Algebra Colloq., 25, 387-398 (2018). · Zbl 1401.13007 · doi:10.1142/S1005386718000287 |
| [21] | D. Bennis and B. Fahid, Rings in which every 2-absorbing ideal is prime, Beitr Algebra Geom, 59, 391-396 (2018) · Zbl 1415.13005 · doi:10.1007/s13366-017-0366-2 |
| [22] | H. Chimal-Dzul and C. A. Lopez-Andrade, When is R[x] a principal ideal ring?, Rev. Integr. Temas Mat., 35(2), 143-148 (2017). · Zbl 1454.13028 · doi:10.18273/revint.v35n2-2017001 |
| [23] | P. J. Cahen, M. Fontana, S. Frisch, and S. Glaz, Open problems in commutative ring theory, Commutative Algebra, Springer, 353-375 (2014). |
| [24] | F. Callialp, E. Yetkin and U. Tekir, On 2-absorbing primary and weakly 2-absorbing elements in multiplicative lattices, Ital. J. Pure Appl. Math., 34, 263-276 (2015). · Zbl 1333.06058 |
| [25] | E. Y. Celikel, E. A. Ugurlu and G. Ulucak, \( On ϕ-2\)-absorbing elements in multiplicative lattices, Palest. J. Math., 5, 127-135 (2016). · Zbl 1359.06010 |
| [26] | E. Y. Celikel, E. A. Ugurlu and G. Ulucak, Onϕ-2-absorbing primary elements in multiplicative lattices, Palest. J. Math., 5, 136-146 (2016). · Zbl 1359.06011 |
| [27] | H. Choi, On n-absorbing ideals of locally divided commutative rings, J. Algebra, 594, 483-518 (2022). · Zbl 1483.13009 · doi:10.1016/j.jalgebra.2021.11.044 |
| [28] | H. Choi and A. Walker, The radical of an \(n\)-absorbing ideal. J. Commut. Algebra, 12(2), 171-177 (2020). · Zbl 1440.13009 · doi:10.1216/jca.2020.12.171 |
| [29] | H. Choi and Andrew Walker, \(n\)-Absorbing monomial ideals in polynomial ring, Int. Electron. J. Algebra, 26, 204-223 (2019). · Zbl 1416.13001 |
| [30] | A.Y. Darani , On 2-absorbing and weakly 2-absorbing ideals of commutative semirings, Kyungpook Math. J., 52, 91-97 (2012). · Zbl 1290.16050 · doi:10.5666/KMJ.2012.52.1.91 |
| [31] | A. Y. Darani and H. Mostafanasab, On 2-absorbing preradicals, J. Algebra Appl., 14, 22 pages (2015). · Zbl 1316.16015 |
| [32] | A.Y. Darani and E.R. Puczylowski, On 2-absorbing commutative semigroups and their applications to rings, Semigroup Forum, 86, 83-91 (2013). · Zbl 1270.20064 · doi:10.1007/s00233-012-9417-z |
| [33] | D. E. Dobbs, Divided rings and going-down, Pacific J. Math., 67(2), 353-363 (1976). · Zbl 0326.13002 · doi:10.2140/pjm.1976.67.353 |
| [34] | G. Donadze, The Anderson-Badawi conjecture for commutative algebras over infinite fields, Indian J. Pure Appl. Math., 47, 691-696 (2016). · Zbl 1368.13002 · doi:10.1007/s13226-016-0194-3 |
| [35] | G.Donadze, A proof of the Anderson-Badawi \(\sqrt{I}^n\subseteq I\) formula for \(n\)-absorbing ideal, Proc. Indian Acad. Sci. Math. Sci., 128(1), paper no. 6, 6 pp (2018). · Zbl 1388.13005 |
| [36] | M. Ebrahimpour and R. Nekooei, On generalizations of prime ideals, Comm. Algebra, 40, 1268-1279 (2012). · Zbl 1278.13003 · doi:10.1080/00927872.2010.550794 |
| [37] | A. El Khalfi, M. Issoual, N. Mahdou and A. Reinhart, Commutative rings with one-absorbing factorization, Comm. Algebra, 49(6), 2689-2703 (2021). · Zbl 1467.13015 · doi:10.1080/00927872.2021.1881105 |
| [38] | A. El Khalfi, N. Mahdou, U. Tekir and S. Koc, On 1-absorbing \(δ\)-primary ideals, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat., Accepted for publication. |
| [39] | J. R. Hedstorm and E. G. Houston, Pseudo-valuation domains, Pacific J. Math., 75, 137-147 (1978). · Zbl 0368.13002 · doi:10.2140/pjm.1978.75.137 |
| [40] | M. Issoual, Rings in which every 2-absorbing primary ideal is primary ideal, Beitr. Algebra Geom., 62, 605-614 (2021). · Zbl 1468.13005 · doi:10.1007/s13366-020-00517-4 |
| [41] | M. Issoual and N. Mahdou, Trivial Extensions defined by 2-absorbing-like conditions, J. Algebra Appl., 17(11), 1850208 (2018). · Zbl 1408.13005 |
| [42] | M. Issoual and N. Mahdou, \( On n\)-AB ring and on \(Ω(R) where R\) is a commutative ring, submitted for publication. |
| [43] | M. Issoual, N. Mahdou and A. M. Moutui, New results about n-absorbing ideals of commutative rings, preprint. |
| [44] | M. Issoual, N. Mahdou and A.M. Moutui, \( On n\)-absorbing and strongly n-absorbing ideals in amalgamated algebras, J. Algebra Appl., 19(10), 2050199 (2019). |
| [45] | M. Issoual, N. Mahdou and A.M. Moutui, \( On (m, n)\)-closed ideals in amalgamated algebras, Int. Electron. J. Algebra 29(2), 134-147 (2021). · Zbl 1474.13041 · doi:10.24330/ieja.852120 |
| [46] | C. Jayaram, U. Tekir and E. Yetkin, 2-absorbing and weakly 2-absorbing elements in multiplicative lattices, Comm. Algebra. 42, 2338-2353 (2014). · Zbl 1302.06027 · doi:10.1080/00927872.2012.761229 |
| [47] | A. Laradji, \( On n\)-absorbing rings and ideals, Colloq. Math., 147, 265-273 (2017). · Zbl 1370.13002 · doi:10.4064/cm6844-5-2016 |
| [48] | A. Malek, A. Hamed and A. Benhissi, 2-absorbing ideals in formal power series rings, Palest. J. Math., 6(2), 502-506 (2017). · Zbl 1371.13003 |
| [49] | H.F.Moghimi and S.R.Naghani, \( On n\)-absorbing ideals and the \(n\)-Krull dimension of a commutative ring, J.Korean Math.Soc., 53(6), 1225-1236 (2016). · Zbl 1355.13005 · doi:10.4134/JKMS.j150072 |
| [50] | M.Muzammil, A.M.Tusif and T.Dumitrescu, Commutative rings with two absorbing factorization, 46(3), 970-978 (2018). · Zbl 1440.13017 |
| [51] | P. Nasehpour, On the Anderson-Badawi \(w_{}R[X](I[X]) = w_{}R(I)\) Conjecture, Arch, Math. Brno. 52, 71-78 (2016). · Zbl 1374.13007 |
| [52] | S. Payrovi ans S. Babaei, On the 2-absorbing ideals, Int. Math. Forum, 7, 265-271 (2012). · Zbl 1247.13016 |
| [53] | S. Smach and S. Hizem, On Anderson-Badawi Conjectures, Beitr. Algebra Geom., 58, 775-785 (2017). · Zbl 1390.13011 · doi:10.1007/s13366-017-0343-9 |
| [54] | M. Tamekkante and E. M. Bouba, (2, \(n)\)-ideals of commutative rings, J. Algebra Appl., 18(6), 1950103 (2019). · Zbl 1412.13005 |
| [55] | U. Tekir, S. Koc and K.H. Oral, \(n\)-Ideals of commutative rings, Filomat, 31(10), 2933-2941 (2017). · Zbl 1488.13016 · doi:10.2298/FIL1710933T |
| [56] | A. Yassine, M.J. Nikmehr and R. Nikandish, On 1-absorbing prime ideals of commutative rings, J. Algebra Appl., 20(10), 2150175 (2021). · Zbl 1479.13006 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
