Abstract
In this article, we study those rings in which every 2-absorbing ideal is prime. We investigate the stability of this property under homorphic image and localization, and its transfer to various contexts of constructions such as direct products, trivial ring extensions and amalgamated algebra along an ideal.
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References
Anderson, D.D., Winders, M.: Idealization of a module. J. Commut. Algebra 1(1), 3–56 (2009)
Anderson, D.F., Badawi, A.: On n-absorbing ideals of commutative rings. Commun. Algebra 39, 1646–1672 (2011)
Badawi, A.: On 2-absorbing ideals of commutative rings. Bull. Aust. Math. Soc. 75, 417–429 (2007)
Badawi, A., Tekir, U., Yeikin, E.: On 2-absorbing primary ideals of commutative rings. Bull. Korean Math. Soc. 51, 1163–1173 (2014)
Badawi, A., Tekir, U., Yeikin, E.: On weakly 2-absorbing primary ideals of commutative rings. J. Korean Math. Soc. 52, 97–111 (2015)
Badawi, A., Tekir, U., Yeikin, E.: Generalizations of 2-absorbing primary ideals of commutative rings. Turk. J. Math. 40, 703–717 (2016)
Badawi, A., Issoual, M., Mahdou, N.: On n-absorbing ideals and (m, n)-closed ideals in trivial ring extension of commutative rings. J. Algebra Appl. 1950123 (2019), 19 pp
Bakkari, C., Kabbaj, S., Mahdou, N.: Trivial extension defined by Prûfer conditions. J. Pure Appl. Algebra 214, 53–60 (2010)
Becker, A.E.: Results on \(n\)-absorbing ideals of commutative rings, M.S. thesis, University of Wisconsin-Milwaukee, Milwaukee, U.S.A. (2015)
Bennis, C., Fahid, B.: Rings in which every 2-absorbing ideal is prime. Beitr. Algebra Geom. 1–6 (2017)
Boisen, M.B., Sheldon, P.B.: CPI-extension: over rings of integral domains with special prime spectrum. Can. J. Math. 29, 722–737 (1977)
Bouba, E.M., Mahdou, N., Tamekkante, M.: Duplication of a module along an ideal. Acta Math. Hung. 154(1), 29–42 (2018)
Callialp, F., Yetkin, E., Tekir, U.: On 2-absorbing primary and weakly 2-absorbing elements in multiplicative lattices. Ital. J. Pure Appl. Math. 34, 263–276 (2015)
Celikel, E.Y., Ugurlu, E.A., Ulucak, G.: On \(\phi \)-2-absorbing elements in multiplicative lattices. Palest. J. Math. 5, 127–135 (2016)
Celikel, E.Y., Ugurlu, E.A., Ulucak, G.: On \(\phi \)-2-absorbing primary elements in multiplicative lattices. Palest. J. Math. 5, 136–146 (2016)
D’Anna, M., Fontana, M.: An amalgamated duplication of a ring along an ideal: the basic properties. J. Algebra Appl. 6, 443–459 (2007)
D’Anna, M., Fontana, M.: The amalgamated duplication of a ring along a multiplicative-canonical ideal. Ark. Mat. 45(2), 241–252 (2007)
D’Anna, M., Finocchiaro, C.A., Fontana, M.: Amalgamated algebra along an ideal. Commutative Algebra and Applications, pp. 155–172. Walter De Gruyter, Boston (2009)
D’Anna, M., Finocchiaro, C.A., Fontana, M.: Properties of chains of prime ideals in amalgamated algebras along an ideal. J. Pure Appl. Algebra 214, 1633–1641 (2010)
D’Anna, M., Finocchiaro, C.A., Fontana, M.: New algebraic properties of an amalgamated algebra along an ideal. Commun. Algebra 44, 1836–1851 (2016)
Darani, A.Y.: On 2-absorbing and weakly 2-absorbing ideals of commutative semirings. Kyungpook Math. J. 52, 91–97 (2012)
Darani, A.Y., Mostafanasab, H.: Co-2-absorbing preradicals and submodules. J. Algebra Appl. 14 (2015), 23 pp
Darani, A.Y., Mostafanasab, H.: On 2-absorbing preradicals. J. Algebra Appl. 14 (2015), 22 pp
Darani, A.Y., Puczylowski, E.R.: On 2-absorbing commutative semigroups and their applications to rings. Semigroup Forum 86, 83–91 (2013)
Gilmer, R.: Ring in which every semi-primary ideals are primary. Pac. J. Math. 12(4) (1962)
Glaz, S.: Commutative Coherent Rings. Lecture Notes in Mathematics, pp. 13–71. Springer, Berlin (1989)
Huckaba, J.A.: Commutative Coherent Rings with Zero Divisors. Marcel Dekker, New York (1988)
Issoual, M., Mahdou, N.: Trivial extensions defined by 2-absorbing-like conditions. J. Algebra Appl. 1850208 (2018), 10 pp
Jayaram, C., Tekir, U., Yetkin, E.: 2-absorbing and weakly 2-absorbing elements in multiplicative lattices. Commun. Algebra 42, 2338–2353 (2014)
Kabbaj, S., Mahdou, N.: Trivial extensions defined by coherent-like conditions. Commun. Algebra 32(10), 3937–3953 (2004)
Kumar, P., Dubey, M.K., Sarohe, P.: Some results on 2-absorbing ideals in commutative semirings. J. Math. Appl. 38, 5–13 (2015)
Kumar, P., Dubey, M.K., Sarohe, P.: On 2-absorbing primary ideals in commutative semirings. Eur. J. Pure Appl. Math. 9, 186–195 (2016)
Laradji, A.: On \(n\)-absorbing rings and ideals. Colloq. Math. 147, 265–273 (2017)
Malek, A., Hamed, A., Benhissi, A.: 2-absorbing ideals in formal power series rings. Palest. J. Math. 6(2), 502–506 (2017)
Nagata, M.: Local Rings. Wiley-Interscience, New York (1962)
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The authors would like to express their sincere thanks for the referee for his/her helpful suggestions and comments.
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Issoual, M., Mahdou, N. (2020). Rings in Which Every 2-Absorbing Ideal Is Prime. In: Shahid, M., Ashraf, M., Al-Solamy, F., Kimura, Y., Vilcu, G. (eds) Differential Geometry, Algebra, and Analysis. ICDGAA 2016. Springer Proceedings in Mathematics & Statistics, vol 327. Springer, Singapore. https://doi.org/10.1007/978-981-15-5455-1_11
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DOI: https://doi.org/10.1007/978-981-15-5455-1_11
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