On weakly 2-absorbing ideal of commutative rings

By Hojjat Mostafanasab, F. Soheilnia, Ahmad Yousefian Darani. 2016

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Let R be a commutative ring with identity 1 6 0. Various generalizations of prime ideals have been studied. For example, a proper ideal I of R isweakly prime if a;b2 R with 06 ab2 I, then either a2 I or b2 I. Also a proper ideal I of R is said to be 2-absorbing if whenever a;b;c 2 R and abc 2 I, then either ab 2 I or ac 2 I or bc 2 I. In this paper, we introduce the concept of a weakly 2-absorbing ideal. A proper ideal I of R is called a weakly 2-absorbing ideal of R if whenever a;b;c2 R and 0 6 abc 2 I, then either ab 2 I or ac 2 I or bc 2 I. For example, every proper ideal of a quasi-local ring (R;M) with M 3 =f0g is a weakly 2-absorbing ideal of R. We show that a weakly 2-absorbing ideal I of R with I 3 6 0 is a 2-absorbing ideal of R. We show that every proper ideal of a commutative ring R is a weakly 2-absorbing ideal if and only if either R is a quasi-local ring with maximal ideal M such that M 3 =f0g or R is ring- isomorphic to R1 F where R1 is a quasi-local ring with maximal ideal M such that M 2 =f0g and F is a eld or R is ring-isomorphic to F1 F2 F3 for some elds F1;F2;F3.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3f65c3af2974bde7ba37388fc1f7e47ed669d523

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Citations in the corpus

On 2-Absorbing Submodules (2012, background): “We defined a submodule N of M is 2-absorbing whenever abm ∈ N for some a, b ∈ R, m ∈ M , then am ∈ N or bm ∈ N or ab ∈ N :R M , see for instance [1, 3, 4, 6, 7, 9, 10].”