On Generalization of \(\phi\)-\(n\)-Absorbing Ideals in Commutative Rings

Pairote Yiarayong, Manoj Siripitukdet

Abstract


The aim of this paper is to extend the concept of \(n\)-absorbing, quasi-\(n\)-absorbing and \(\phi\)-\(n\)-absorbing ideals given by  Anderson and Badawi [2] to the context of \(\phi\)-semi-\(n\)-absorbing ideals. Let \(\phi:\mathcal{I}(R)\rightarrow \mathcal{I}(R)\cup \left\lbrace \emptyset \right \rbrace\) be a function where \(\mathcal{I}(R)\) is the set of all ideals of \(R\). A proper ideal \(R\) of \(R\) is called a \(\phi\)-semi-\(n\)-absorbing Ideal, if for each \(a \in R\) with \(a^{n+1} \in I - \phi(I)\), then \(a^{n}\in I\). Some characterizations of semi-\(n\)-absorbing ideals are obtained.\ It is shown that if \(J\) is a \(\phi\)-semi-\(n\)-absorbing ideal of \(R\), then \(J/I\) is a \(\phi_{I}\)-semi-\(n\)-absorbing ideal of \(R/I\) where \(I \subseteq J\). A number of results concerning relationships between \(\phi\)-semi-\(n\)-absorbing, \(\phi_{0}\)-semi-\(n\)-absorbing, \(\phi_{\emptyset}\)-semi-2-absorbing and \(\phi_{n\geq 2}\)-semi-\(n\)-absorbing ideals of commutative rings. Finally, we obtain sufficient conditions of a semi-\(n\)-absorbing ideal in order to be a \(\phi\)-semi-\(n\)-absorbing ideal.

Keywords


\(n\)-absorbing ideal; Quasi-\(n\)-absorbing; \(\phi\)-\(n\)-absorbing ideal; Semi-\(n\)-absorbing ideal; \(\phi\)-semi-\(n\)-absorbing ideal

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References


D.F. Anderson and A. Badawi, On n-absorbing ideals of commutative rings, Comm. Algebra 39(5) (2011), 1646 – 1672.

D.F. Anderson and A. Badawi, On (m,n)-closed ideals of commutative rings, Journal of Algebra and Its Applications 16(1) (2017), 1750013 (21 pages).

D.D. Anderson and M. Bataineh, Generalizations of prime ideals, Comm. Algebra 36 (2008), 686 – 696.

A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Aust. Math. Soc. 75(2007), 417 – 429.

A. Badawi and A.Y. Darani, On weakly 2-absorbing ideals of commutative rings, Houston J. Math. 39(2013), 441 – 452.

M. Ebrahimpour and R. Nekooei, On generalizations of prime ideals, Comm. Algebra 40(2012), 1268 – 1279.

H. Mostafanasab and A.Y. Darani, On n-absorbing ideals and two generalizations of semiprime ideals, Thai J. Math. 15(2) (2017), 387 – 408.

H. Mostafanasab, F. Soheilnia and A.Y. Darani, On weakly n-absorbing ideals of commutative rings, An. S¸ tiint¸. Univ. Al. I. Cuza Ias¸i Mat. (N.S.) 3(f.2) (2016), 845 – 862.


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