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On almost radical ideals of noncommutative rings | ||
| Journal of Algebra and Related Topics | ||
| دوره 12، شماره 2، فروردین 2025، صفحه 25-36 اصل مقاله (282.31 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2024.22580.1413 | ||
| نویسنده | ||
| N. J. Groenewald* | ||
| Department of Mathematics, Nelson Mandela University, Gqeberha, South Africa | ||
| چکیده | ||
| Let $\mathcal{J}(R)$ denote the Jacobson radical of a commutative ring $R$. In \cite{Khashan} the notion of a $\mathcal{J}$-ideal was introduced and studied. In \cite{khashanandece} Khashan et al. introduced the concept of weakly $\mathcal{J}$-ideals as a new generalization of $\mathcal{J}$-ideals. In \cite{groenewald} and \cite{groenerwaldweakly} it was shown that many of the results are special cases of a more general situation. In \cite{TekirEce} the notion of an almost n-ideal was introduced and studied. In this note, we define almost $\rho $-ideals for a special radical $\rho $. If $\rho $ is the prime radical then we have the almost n-ideals for noncommutative rings. We prove amongst other results that an ideal $I$ of a ring is an almost $\rho $-ideal if and only if $I/I^{2}$ is a weakly $\rho $-ideal of $R.$ We also investigate rings for which every ideal is an almost radical ideal for a special radical $\rho $. | ||
| کلیدواژهها | ||
| Special radical $\rho $؛ $\rho $-ideal؛ Weakly-$\rho $-ideal؛ Almost-$\rho $-ideal؛ Fully almost-$\rho $-ring | ||
| مراجع | ||
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[1] E. Y. Celikel, On generalization of n-ideals, Erzincan Universitesi Fen Bilimleri Enstitusu Dergisi, 12 (2019), 650-659. [2] B. de la Rosa and S. Veldsman, A relationship between ring radicals and module radicals, Quaest. Math., 17 (1994), 453-467. [3] B. J. Gardner, R. Wiegandt, Radical Theory of Rings, Marcel Dekker Inc, New York, 2004. [4] N. J. Groenewald, On radical ideals of noncommutative rings, J. Algebra Appl., (09) 22 (2023), 2350196. [5] N.J. Groenewald, On weakly radical ideals of noncommutative rings, J. Algebra Appl., (12) 22 (2023), 2330003. [6] N.J. Groenewald, Generalizations of radical ideals in noncommutative rings, to appear in J. Algebr. Syst. [7] H. A. Khashan and E. Y. Celikel, Weakly J-ideals of commutative rings, Filomat, (2) 36 (2022), 485-495. [8] H. A. Khashan and A. B. Bani-Ata, J-ideals of commutative rings, Int. Electron. J. Algebra, 29 (2021), 148-164. [9] T. Y. Lam, A First Course in Noncommutative Rings, second ed., Graduate Texts in Mathematics, 131, Springer-Verlag, New York, 2001. [10] U. Tekir, S. Koc and K.H. Oral, n-Ideals of commutative rings, Filomat, (10) 31 (2017), 2933-2941. [11] S. Veldsman, A note on the radicals of idealizations, Southeast Asian Bull. Math., 32 (2008), 545-551. | ||
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