# ON WEAKLY 2-ABSORBING PRIMARY IDEALS OF COMMUTATIVE RINGS

• Tekir, Unsal ;
• Yetkin, Ece
• Published : 2015.01.01
• 93 8

#### Abstract

Let R be a commutative ring with $1{\neq}0$. In this paper, we introduce the concept of weakly 2-absorbing primary ideal which is a generalization of weakly 2-absorbing ideal. A proper ideal I of R is called a weakly 2-absorbing primary ideal of R if whenever a, b, $c{\in}R$ and $0{\neq}abc{\in}I$, then $ab{\in}I$ or $ac{\in}\sqrt{I}$ or $bc{\in}\sqrt{I}$. A number of results concerning weakly 2-absorbing primary ideals and examples of weakly 2-absorbing primary ideals are given.

#### Keywords

primary ideal;weakly primary ideal;prime ideal;weakly prime ideal;2-absorbing ideal;n-absorbing ideal;weakly 2-absorbing ideal;2-absorbing primary ideal;weakly 2-absorbing primary ideal

#### References

1. D. D. Anderson and M. Bataineh, Generalizations of prime ideals, Comm. Algebra 36 (2008), no. 2, 686-696. https://doi.org/10.1080/00927870701724177
2. A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc. 75 (2007), no. 3, 417-429. https://doi.org/10.1017/S0004972700039344
3. D. D. Anderson and E. Smith, Weakly prime ideals, Houston J. Math. 29 (2003), no. 4, 831-840.
4. D. F. Anderson and A. Badawi, On n-absorbing ideals of commutative rings, Comm. Algebra 39 (2011), no. 5, 1646-1672. https://doi.org/10.1080/00927871003738998
5. S. Ebrahimi Atani and F. Farzalipour, On weakly primary ideals, Georgian Math. J. 12 (2005), no. 3, 423-429.
6. A. Badawi and A. Y. Darani, On weakly 2-absorbing ideals of commutative rings, Houston J. Math. 39 (2013), no. 2, 441-452.
7. A. Badawi, U. Tekir, and E. Yetkin, On 2-absorbing primary ideals in commutativerings, Bull. Korean Math. Soc. (in press)
8. A. Y. Darani and E. R. Puczylowski, On 2-absorbing commutative semigroups and their applications to rings, Semigroup Forum 86 (2013), no. 1, 83-91. https://doi.org/10.1007/s00233-012-9417-z
9. M. Ebrahimpour and R. Nekooei, On generalizations of prime ideals, Comm. Algebra 40 (2012), no. 4, 1268-1279. https://doi.org/10.1080/00927872.2010.550794
10. R. Gilmer, Multiplicative Ideal Theory, Queen's Papers Pure Appl. Math. 90, Queen's University, Kingston, 1992.
11. J. Huckaba, Rings with Zero-Divisors, New York/Basil: Marcel Dekker, 1988.
12. S. Payrovi and S. Babaei, On the 2-absorbing ideals, Int. Math. Forum 7 (2012), no. 5-8, 265-271.

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