On weakly 2-absorbing δ-primary ideals of commutative rings

Ayman Badawi 1  and Brahim Fahid 2
  • 1 Department of Mathematics & Statistics, American University of Sharjah, P.O. Box 26666, Sharjah, United Arab Emirates
  • 2 Department of Mathematics, Mohammed V University, Faculty of Sciences, B.P. 1014, Rabat, Morocco
Ayman Badawi
  • Corresponding author
  • Department of Mathematics & Statistics, American University of Sharjah, P.O. Box 26666, Sharjah, United Arab Emirates
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and Brahim Fahid

Abstract

Let R be a commutative ring with 10. We recall that a proper ideal I of R is called a weakly 2-absorbing primary ideal of R if whenever a,b,cR and 0abcI, then abI or acI or bcI. In this paper, we introduce a new class of ideals that is closely related to the class of weakly 2-absorbing primary ideals. Let I(R) be the set of all ideals of R and let δ:I(R)I(R) be a function. Then δ is called an expansion function of ideals of R if whenever L,I,J are ideals of R with JI, then Lδ(L) and δ(J)δ(I). Let δ be an expansion function of ideals of R. Then a proper ideal I of R (i.e., IR) is called a weakly 2-absorbing δ-primary ideal if 0abcI implies abI or acδ(I) or bcδ(I). For example, let δ:I(R)I(R) such that δ(I)=I. Then δ is an expansion function of ideals of R, and hence a proper ideal I of R is a weakly 2-absorbing primary ideal of R if and only if I is a weakly 2-absorbing δ-primary ideal of R. A number of results concerning weakly 2-absorbing δ-primary ideals and examples of weakly 2-absorbing δ-primary ideals are given.

  • [1]

    D. D. Anderson and E. Smith, Weakly prime ideals, Houston J. Math. 29 (2003), no. 4, 831–840.

  • [2]

    D. F. Anderson and A. Badawi, On n-absorbing ideals of commutative rings, Comm. Algebra 39 (2011), no. 5, 1646–1672.

    • Crossref
    • Export Citation
  • [3]

    D. F. Anderson and A. Badawi, On ( m , n ) (m,n)-closed ideals of commutative rings, J. Algebra Appl. 16 (2017), no. 1, Article ID 1750013.

  • [4]

    A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Aust. Math. Soc. 75 (2007), no. 3, 417–429.

    • Crossref
    • Export Citation
  • [5]

    A. Badawi, On weakly semiprime ideals of commutative rings, Beitr. Algebra Geom. 57 (2016), no. 3, 589–597.

    • Crossref
    • Export Citation
  • [6]

    A. Badawi, U. Tekir and E. Yetkin, On 2-absorbing primary ideals in commutative rings, Bull. Korean Math. Soc. 51 (2014), no. 4, 1163–1173.

    • Crossref
    • Export Citation
  • [7]

    A. Badawi, U. Tekir and E. Yetkin, On weakly 2-absorbing primary ideals of commutative rings, J. Korean Math. Soc. 52 (2015), no. 1, 97–111.

    • Crossref
    • Export Citation
  • [8]

    A. Badawi and A. Yousefian Darani, On weakly 2-absorbing ideals of commutative rings, Houston J. Math. 39 (2013), no. 2, 441–452.

  • [9]

    A. Y. Darani and H. Mostafanasab, On 2-absorbing preradicals, J. Algebra Appl. 14 (2015), no. 2, Article ID 1550017.

  • [10]

    S. Ebrahimi Atani and F. Farzalipour, On weakly primary ideals, Georgian Math. J. 12 (2005), no. 3, 423–429.

  • [11]

    B. Fahid and Z. Dongsheng, 2-absorbing δ-primary ideals in commutative rings, Kyungpook Math. J. 57 (2017), no. 2, 193–198.

  • [12]

    C. Huneke and I. Swanson, Integral Closure of Ideals, Rings, and Modules, London Math. Soc. Lecture Note Ser. 336, Cambridge University Press, Cambridge, 2006.

  • [13]

    H. Mostafanasab, U. Tekir and K. Hakan Oral, Classical 2-absorbing submodules of modules over commutative rings, Eur. J. Pure Appl. Math. 8 (2015), no. 3, 417–430.

  • [14]

    S. Payrovi and S. Babaei, On the 2-absorbing ideals, Int. Math. Forum 7 (2012), no. 5–8, 265–271.

  • [15]

    A. Yousefian Darani and H. Mostafanasab, Co-2-absorbing preradicals and submodules, J. Algebra Appl. 14 (2015), no. 7, Article ID 1550113.

  • [16]

    A. Yousefian Darani and E. R. Puczył owski, On 2-absorbing commutative semigroups and their applications to rings, Semigroup Forum 86 (2013), no. 1, 83–91.

    • Crossref
    • Export Citation
  • [17]

    D. Zhao, δ-primary ideals of commutative rings, Kyungpook Math. J. 41 (2001), no. 1, 17–22.

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