IJPAM: Volume 99, No. 1 (2015)

$\varphi$-$2$-ABSORBING IDEALS

A. Khaksari
Department of Mathematics
Payame Noor University
P.O. Box: 19395-3697, Tehran, IRAN

Abstract. Let $R$ be a commutative ring with identity. $2$-absorbing ideals have been studied by A. Badawi. A proper ideal $I$ of $R$ is $2$-absorbing if $a,b,c \in R$ with $abc\in I$ implies $ab\in I$ or $ac \in I$ or $bc\in I$. Let $\varphi: I(R) \rightarrow
I(R)\cup \{\emptyset \}$ be a function where $I(R)$ is the set of ideals of $R$. We call a proper ideal $I$ of $R$ a $\varphi$-2-absorbing ideal if $a,b,c \in R$ with $abc\in
I-\varphi(I)$ implies $ab\in I$ or $ac \in I$ or $bc\in I$. So taking $\varphi_\emptyset(J)= \emptyset$ (resp., $\varphi_0(J)=0,
\varphi_2(J)=J^2$), a $\varphi_{\emptyset}$-$2$-absorbing ideal (resp., $\varphi_0$-$2$-absorbing ideal, $\varphi_2$- $2$-absorbing ideal) is a $2$-absorbing ideal (resp., weakly $2$-absorbing ideal, almost $2$-absorbing ideal). We show that $\varphi$-$2$-absorbing ideals enjoy analogs of many of the properties of $2$-absorbing ideals.

Received: February 22, 2014

AMS Subject Classification: 13A15

Key Words and Phrases: almost $2$-absorbing ideal, $2$-absorbing ideal, $\varphi$-$2$-absorbing ideal, weakly $2$-absorbing ideal

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DOI: 10.12732/ijpam.v99i1.1 How to cite this paper?

International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2015
Volume: 99
Issue: 1
Pages: 1 -

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