J-IDEALS OF COMMUTATIVE RINGS
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,
___
- 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.
- ISSN: 1306-6048
- Yayın Aralığı: Yılda 2 Sayı
- Başlangıç: 2007
- Yayıncı: Abdullah HARMANCI
Sayıdaki Diğer Makaleler
F. SOHEILNIA, Sh. PAYROVI, A. BEHTOEI
Mohammad JAVAHERI, Nikolai A. KRYLOV
Ayman M. A. HOROUB, W. K. NICHOLSON
Ghorban SOLEYMANPOUR, Ali S. JANFADA
Gabriel PICAVET, Martine PICAVET-L'HERMITTE
Mohammed ISSOUAL, Najib MAHDOU, Moutu Abdou Salam MOUTUI