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

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.

Keywords

• Graded prime ideal,graded 2-absorbing ideal,graded weakly 2-absorbing ideal

References

1. [1] D. D. Anderson and M. Bataineh, Generalizations of Prime Ideals, Communications in Algebra, 36 (2008), 686–696.
2. [2] D. D .Anderson and E. Smith, Weakly Prime Ideals, Houston Journal of Mathematics, 29 (2003), 831–840.
3. [3] D. F. Anderson and A. Badawi, On n-absorbing Ideals of Commutative Rings, Communications in Algebra, 39 (2011), 1646–1672.
4. [4] S. E. Atani, On Graded Weakly Prime Ideals, Turkish Journal of Mathematics, 30 (2006), 351–358.
5. [5] A. Badawi, On 2-absorbing Ideals of Commutative Rings, Bulletine of the Australian Mathematical Society, 75 (2007), 417–429.
6. [6] A. Badawi and A. Y. Darani, On Weakly 2-absorbing Ideals of Commutative Rings, Houston Journal of Mathematics, (in press).
7. [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. [8] M. Ebrahimpour and R. Nekooei, On Generalizations of Prime Ideals, Communications in Algebra, 40, (2012), 1268–1279.

9. [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. [10] J. Huckaba, Rings with Zero-Divisors, New York/Basil: Marcel Dekker, (1988).
11. [11] S. Moradi and A. Azizi, 2-Absorbing and n-weakly Prime Submodules, Miskolc Mathematical Notes, 13, (2012), 75–86.
12. [12] C. Nastasescu and F. Van Oystaeyen, Graded Ring Theory, Mathematical Library 28, Amsterdam: North Holland publishing company, (1982).
13. [13] D. G. Northcott, On Homogeneous Ideals, Proceedings of the Glasgow Mathematical Association, 2, (1955), 105–111.
14. [14] M. Refai and K. Al-Zoubi, On Graded Primary Ideals, Turkish Journal of Mathematics, 28, (2004), 217–229.