On almost radical ideals of noncommutative rings

Document Type : Research Paper

Author

Department of Mathematics, Nelson Mandela University, Gqeberha, South Africa

Abstract

‎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 $‎.

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References

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Journal of Algebra and Related Topics
Volume 12, Issue 2
January 2025
Pages 25-36

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