On 2-Absorbing Quasi-Primary Ideals in Commutative Rings
- PDF / 404,674 Bytes
- 8 Pages / 439.37 x 666.142 pts Page_size
- 37 Downloads / 203 Views
On 2-Absorbing Quasi-Primary Ideals in Commutative Rings Unsal Tekir1 · Suat Koç1 · Kursat Hakan Oral2 · Kar Ping Shum3
Received: 23 June 2015 / Revised: 16 October 2015 / Accepted: 19 October 2015 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2015
Abstract Let R be a commutative ring with nonzero identity. In this article, we introduce the notion of 2-absorbing quasi-primary ideal which is a generalization of quasi-primary ideal. We define a proper ideal I of R to be 2-absorbing quasi primary √ if I is a 2-absorbing ideal of R. A number of results concerning 2-absorbing quasiprimary ideals and examples of 2-absorbing quasi-primary ideals are given. Keywords
Prime ideal · 2-Absorbing ideal · 2-Absorbing quasi-primary ideal
Mathematics Subject Classification
13A15 · 13F05 · 13G05
1 Introduction In this paper all rings are commutative with nonzero identity. Let R be a ring. Let I be a proper ideal of R, the set {r ∈ R | r s ∈ I for some s ∈ R\I } will be denoted by
B
Unsal Tekir [email protected] Suat Koç [email protected] Kursat Hakan Oral [email protected] Kar Ping Shum [email protected]
1
Department of Mathematics, Marmara University, Istanbul, Turkey
2
Department of Mathematics, Yildiz Technical University, Istanbul, Turkey
3
Institute of Mathematics, Yunnan University, Kunming 650091, People’s Republic of China
123
U. Tekir et al.
√ Z I (R). Also the radical of I is defined as I := {r ∈ R | r k ∈ I for some k ∈ N} and for x ∈ R, (I : x) denote the ideal {r ∈ R | r x ∈ I } of R. Quasi-primary ideals in commutative rings were introduced √and investigated by Fuchs in [1]. An ideal I of R is called a quasi-primary ideal if I is a prime ideal. The notion of 2-absorbing ideal, which is a generalization of prime ideal, was introduced by Badawi in [2] and investigated in [3–7]. Various generalizations of prime ideals are studied in [8,9]. Also the notion of 2-absorbing primary ideal, which is a generalization of primary ideal, was introduced by Badawi, Tekir and Yetkin in [10]. A proper ideal I of R is called a 2-absorbing ideal of R if whenever a, b, c ∈ R and abc ∈ I , then ab ∈ I or ac ∈ I or bc ∈ I . Also a proper ideal I of R is called a 2-absorbing primary √ ideal of R if √ whenever a, b, c ∈ R and abc ∈ I , then ab ∈ I or ac ∈ I or bc ∈ I . Note that a 2-absorbing ideal of a commutative ring R is a 2-absorbing primary ideal of R. But the converse is not true. For example, consider the ideal I = (20) of Z. Since 2.2.5 ∈ I , but 2.2 ∈ / I and 2.5 ∈ / I , I is not a 2-absorbing ideal √ of Z. However, it is clear that I is a 2-absorbing primary ideal of Z, since 2.5 ∈ I . In this paper, we introduce the notion of 2-absorbing quasi-primary ideal which is a generalization of quasi-primary √ ideal. A proper ideal I of R is called a 2-absorbing quasi-primary ideal of R if I is a 2-absorbing ideal of R. It is clear that every 2-absorbing primary ideal of a ring R is a 2-absorbing quasi-primary ideal of R from [10, Theorem
Data Loading...
