Abstract
Let R be a commutative ring with 1 ≠ 0 and U(R) be the set of all unit elements of R. Let m, n be positive integers such that m > n. In this article, we study a generalization of n-absorbing ideals. A proper ideal I of R is called an (m, n)-absorbing ideal if whenever a 1⋯a m ∈I for a 1,…, a m ∈R∖U(R), then there are n of the a i ’s whose product is in I. We investigate the stability of (m, n)-absorbing ideals with respect to various ring theoretic constructions and study (m, n)-absorbing ideals in several commutative rings. For example, in a Bézout ring or a Boolean ring, an ideal is an (m, n)-absorbing ideal if and only if it is an n-absorbing ideal, and in an almost Dedekind domain every (m, n)-absorbing ideal is a product of at most m − 1 maximal ideals.
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References
Anderson D F and Badawi A, On n-absorbing ideal in commutative rings, Comm. Algebra 39 (2011) 1646–1672
Anderson D D and Winders M, Idealization of a module, Comm. Algebra 1 (2009) 3–56
Badawi A, On divided commutative rings, Comm. Algebra 27 (1999) 1465–1474
Badawi A, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc. 75 (2007) 417–429
Badawi A and Darani A Y, On weakly 2-absorbing ideals of commutative rings, Houston J. Math. 39 (2013) 441–452
Badawi A, Tekir U and Yetkin E, On 2-absorbing primary ideals of commutative rings, Bull. Korean Math. Soc. 51 (2014) 1163–1173
Badawi A, Tekir U and Yetkin E, On weakly 2-absorbing primary ideals of commutative rings, J. Korean Math. Soc. 52 (2015) 97–111
Cohn P M, Bézout rings and their subrings, Proc. Camb. Phil. Soc. 64 (1968) 251–264
Darani A Y and Mostafanasab H, On 2-absorbing preradicals, J. Algebra Appl. 14 (2015) 22
Darani A Y and Mostafanasab H, Co-2-absorbing preradicals and submodules, J. Algebra Appl. 14 (2015) 23
Darani A Y and Puczylowski E R, On 2-absorbing commutative semigroups and their applications to rings, Semigroup Forum 86 (2013) 83–91
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JALAL ABADI, B.Z., MOGHIMI, H.F. On (m, n)-absorbing ideals of commutative rings. Proc Math Sci 127, 251–261 (2017). https://doi.org/10.1007/s12044-016-0323-2
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DOI: https://doi.org/10.1007/s12044-016-0323-2