J-IDEALS OF COMMUTATIVE RINGS

Year 2021, Volume: 29 Issue: 29, 148 - 164, 05.01.2021

Hani A. Khashan Amal B. Banı-ata

https://doi.org/10.24330/ieja.852139

Cited By: 10

Abstract

Let $R$ be a commutative ring with identity and $N(R)$ and
$J\left(R\right)$ denote the nilradical and the Jacobson radical
of $R$, respectively. A proper ideal $I$ of $R$ is called an
n-ideal if for every $a,b\in R$, whenever $ab\in I$\ and $a\notin
N(R)$, then $b\in I$. In this paper, we introduce and study
J-ideals as a new generalization of n-ideals in commutative rings.
A proper ideal $I$\ of $R$\ is called a J-ideal if whenever $ab\in
I$\ with $a\notin J\left(R\right) $, then $b\in I$\ for every
$a,b\in R$. We study many properties and examples of such class of
ideals. Moreover, we investigate its relation with some other
classes of ideals such as r-ideals, prime, primary and maximal
ideals. Finally, we, more generally, define and study J-submodules
of an $R$-modules $M$. We clarify some of their properties
especially in the case of multiplication modules.

Keywords

n-ideal, r-ideal, J-ideal, J-submodule

References

• D. D. Anderson and M. Bataineh, Generalizations of prime ideals, Comm. Algebra, 36(2) (2008), 686-696.
• D. D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra, 1(1) (2009), 3-56.
• D. D. Anderson, T. Arabaci, U. Tekir and S. Koc, On S-multiplication modules, Comm. Algebra, 48(8) (2020), 3398-3407.
• D. D. Anderson, M. Axtell, S. J. Forman and J. Stickles, When are associates unit multiples, Rocky Mountain J. Math., 34(3) (2004), 811-828.
• A. Badawi, U. Tekir and E. Yetkin, On 2-absorbing primary ideals in commutative rings, Bull. Korean Math. Soc., 51(4) (2014), 1163-1173.
• S. Ebrahimi Atani and F. Farzalipour, On weakly primary ideals, Georgian Math. J., 12(3) (2005), 423-429.
• M. Ebrahimpour and R. Nekooei, On generalizations of prime ideals, Comm. Algebra, 40(4) (2012), 1268-1279.
• Z. A. El-Bast and P. F. Smith, Multiplication modules, Comm. Algebra, 16(4) (1988), 755-779.
• H. A. Khashan, On almost prime submodules, Acta Math. Sci. Ser. B (Engl. Ed.), 32(2) (2012), 645-651.
• R. Mohamadian, r-Ideals in commutative rings, Turkish J. Math., 39 (2015), 733-749.
• R. Y. Sharp, Steps in Commutative Algebra, Second edition, London Mathematical Society Student Texts, 51, Cambridge University Press, Cambridge, 2000.
• P. F. Smith, Some remarks on multiplication modules, Arch. Math. (Basel), 50(3) (1988), 223-235.
• U. Tekir, S. Koc and K. H. Oral, n-Ideals of commutative rings, Filomat, 31(10) (2017), 2933-2941.