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Year 2016, Volume: 13 Issue: 2, - , 01.11.2016

Abstract

References

  • [1] D. D. Anderson and M. Bataineh, Generalizations of Prime Ideals, Communications in Algebra, 36 (2008), 686–696.
  • [2] D. D .Anderson and E. Smith, Weakly Prime Ideals, Houston Journal of Mathematics, 29 (2003), 831–840.
  • [3] D. F. Anderson and A. Badawi, On n-absorbing Ideals of Commutative Rings, Communications in Algebra, 39 (2011), 1646–1672.
  • [4] S. E. Atani, On Graded Weakly Prime Ideals, Turkish Journal of Mathematics, 30 (2006), 351–358.
  • [5] A. Badawi, On 2-absorbing Ideals of Commutative Rings, Bulletine of the Australian Mathematical Society, 75 (2007), 417–429.
  • [6] A. Badawi and A. Y. Darani, On Weakly 2-absorbing Ideals of Commutative Rings, Houston Journal of Mathematics, (in press).
  • [7] A. Y. Darani and E. R. Puczylowski, On 2-absorbing Commutative Semigroups and their Applications to Rings, Semigroup Forum, 86, (2013), 83–91.
  • [8] M. Ebrahimpour and R. Nekooei, On Generalizations of Prime Ideals, Communications in Algebra, 40, (2012), 1268–1279.
  • [9] H. Fazaeli Moghimi and S. Rahimi Naghani, On n-absorbing Ideals and the n-Krull Dimension of a Commutative Ring, Journal of the Korean Mathematical Society, in press.
  • [10] J. Huckaba, Rings with Zero-Divisors, New York/Basil: Marcel Dekker, (1988).
  • [11] S. Moradi and A. Azizi, 2-Absorbing and n-weakly Prime Submodules, Miskolc Mathematical Notes, 13, (2012), 75–86.
  • [12] C. Nastasescu and F. Van Oystaeyen, Graded Ring Theory, Mathematical Library 28, Amsterdam: North Holland publishing company, (1982).
  • [13] D. G. Northcott, On Homogeneous Ideals, Proceedings of the Glasgow Mathematical Association, 2, (1955), 105–111.
  • [14] M. Refai and K. Al-Zoubi, On Graded Primary Ideals, Turkish Journal of Mathematics, 28, (2004), 217–229.

On Graded 2-Absorbing and Graded Weakly 2-Absorbing Ideals of a Commutative Ring

Year 2016, Volume: 13 Issue: 2, - , 01.11.2016

Abstract

Let G be a group with identity e and R be a G-graded commutative ring with 1 6= 0. In this paper,
we study the graded versions of 2-absorbing and weakly 2-absorbing ideals which are generalizations of the
graded prime and graded weakly prime ideals, respectively. A graded proper ideal I of R is called a graded 2-
absorbing (resp. graded weakly 2-absorbing) ideal if whenever abc ∈ I (resp. 0 != abc ∈ I) for homogeneous
elements a,b, c ∈ R, then ab ∈ I or ac ∈ I or bc ∈ I. It is clear that a graded ideal which is a 2-absorbing
ideal, is a graded 2-absorbing ideal, but we show that the converse is not true in general. It is proved that if
I = ⊕g∈GIg is a graded weakly 2-absorbing ideal of R, then either I is a 2-absorbing ideal of R or I3g = (0) for
all g ∈ G. It is also shown that if I = ⊕i∈GIg is a graded weakly 2-absorbing ideal of R, then for each g ∈ G,
either Ig is a 2-absorbing Re-submodule of Rg or (Ig :Re Rg)
2
Ig = 0.

References

  • [1] D. D. Anderson and M. Bataineh, Generalizations of Prime Ideals, Communications in Algebra, 36 (2008), 686–696.
  • [2] D. D .Anderson and E. Smith, Weakly Prime Ideals, Houston Journal of Mathematics, 29 (2003), 831–840.
  • [3] D. F. Anderson and A. Badawi, On n-absorbing Ideals of Commutative Rings, Communications in Algebra, 39 (2011), 1646–1672.
  • [4] S. E. Atani, On Graded Weakly Prime Ideals, Turkish Journal of Mathematics, 30 (2006), 351–358.
  • [5] A. Badawi, On 2-absorbing Ideals of Commutative Rings, Bulletine of the Australian Mathematical Society, 75 (2007), 417–429.
  • [6] A. Badawi and A. Y. Darani, On Weakly 2-absorbing Ideals of Commutative Rings, Houston Journal of Mathematics, (in press).
  • [7] A. Y. Darani and E. R. Puczylowski, On 2-absorbing Commutative Semigroups and their Applications to Rings, Semigroup Forum, 86, (2013), 83–91.
  • [8] M. Ebrahimpour and R. Nekooei, On Generalizations of Prime Ideals, Communications in Algebra, 40, (2012), 1268–1279.
  • [9] H. Fazaeli Moghimi and S. Rahimi Naghani, On n-absorbing Ideals and the n-Krull Dimension of a Commutative Ring, Journal of the Korean Mathematical Society, in press.
  • [10] J. Huckaba, Rings with Zero-Divisors, New York/Basil: Marcel Dekker, (1988).
  • [11] S. Moradi and A. Azizi, 2-Absorbing and n-weakly Prime Submodules, Miskolc Mathematical Notes, 13, (2012), 75–86.
  • [12] C. Nastasescu and F. Van Oystaeyen, Graded Ring Theory, Mathematical Library 28, Amsterdam: North Holland publishing company, (1982).
  • [13] D. G. Northcott, On Homogeneous Ideals, Proceedings of the Glasgow Mathematical Association, 2, (1955), 105–111.
  • [14] M. Refai and K. Al-Zoubi, On Graded Primary Ideals, Turkish Journal of Mathematics, 28, (2004), 217–229.
There are 14 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

Sadegh Rahimi Naghani This is me

Hosein Fazaeli Moghimi

Publication Date November 1, 2016
Published in Issue Year 2016 Volume: 13 Issue: 2

Cite

APA Naghani, S. R., & Moghimi, H. F. (2016). On Graded 2-Absorbing and Graded Weakly 2-Absorbing Ideals of a Commutative Ring. Cankaya University Journal of Science and Engineering, 13(2).
AMA Naghani SR, Moghimi HF. On Graded 2-Absorbing and Graded Weakly 2-Absorbing Ideals of a Commutative Ring. CUJSE. November 2016;13(2).
Chicago Naghani, Sadegh Rahimi, and Hosein Fazaeli Moghimi. “On Graded 2-Absorbing and Graded Weakly 2-Absorbing Ideals of a Commutative Ring”. Cankaya University Journal of Science and Engineering 13, no. 2 (November 2016).
EndNote Naghani SR, Moghimi HF (November 1, 2016) On Graded 2-Absorbing and Graded Weakly 2-Absorbing Ideals of a Commutative Ring. Cankaya University Journal of Science and Engineering 13 2
IEEE S. R. Naghani and H. F. Moghimi, “On Graded 2-Absorbing and Graded Weakly 2-Absorbing Ideals of a Commutative Ring”, CUJSE, vol. 13, no. 2, 2016.
ISNAD Naghani, Sadegh Rahimi - Moghimi, Hosein Fazaeli. “On Graded 2-Absorbing and Graded Weakly 2-Absorbing Ideals of a Commutative Ring”. Cankaya University Journal of Science and Engineering 13/2 (November 2016).
JAMA Naghani SR, Moghimi HF. On Graded 2-Absorbing and Graded Weakly 2-Absorbing Ideals of a Commutative Ring. CUJSE. 2016;13.
MLA Naghani, Sadegh Rahimi and Hosein Fazaeli Moghimi. “On Graded 2-Absorbing and Graded Weakly 2-Absorbing Ideals of a Commutative Ring”. Cankaya University Journal of Science and Engineering, vol. 13, no. 2, 2016.
Vancouver Naghani SR, Moghimi HF. On Graded 2-Absorbing and Graded Weakly 2-Absorbing Ideals of a Commutative Ring. CUJSE. 2016;13(2).