On semi radical ideals of noncommutative rings
Nico Groenewald
Department of Mathematics, Nelson Mandela University, Port Elizabeth, South Africa.
Pages 23-35 | Received 01 July 2023, Accepted 21 September 2023, Published 30 June 2024
Abstract
Since the introduction of n-ideals and J-ideals in commutative rings many different aspects of these ideals have been investigated. As a generalization the notion of weakly n-ideals and weakly J-ideals was introduced and studied. Recently it was proved that many of the results are also true for noncommutative rings as a special case of a more general situation. In a recent paper Khashan et. al introduced the notion of semi n-ideals as a generalization of n-ideals where n is the prime radical and studied this generalization. In this note we show that these results are special cases of a more general situation. If \rho is a special radical and R a noncommutative ring then the ideal I of R is a semi \rho-ideal if aRa \subseteq I, then a \in \rho (R) or a \in I. This covers a wide spectrum of semi ideals and if \rho is the prime radical we have the notion of semi n-ideals for noncommutative rings. In this note we prove that most of the results for the semi n-ideals are satisfied for noncommutative rings as a special case.
Keywords: Special radical, semi n-ideal, semi \rho-ideal, semi \rho-submodule.
MSC numbers: 16N20, 16N40, 16N80, 16L30.
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