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1-absorbing prime ideals and weakly 1-absorbing prime ideals in noncommutative rings

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Abstract

This paper introduce and study 1-absorbing prime ideals and weakly 1-absorbing prime ideals in noncommutative rings a new class of ideals between prime ideals (weakly prime ideals) and 2-absorbing ideals (weakly 2-absorbing ideals). Let R be a noncommutative ring with a nonzero identity \(1\ne 0\). A proper ideal P of R is said to be a 1-absorbing prime ideal (weakly 1-absorbing prime ideal) if for all nonunits \(x,y,z\in R\) with xRyRz \(\subset \) P \((\left\{ 0\right\} \ne \) xRyRz \(\subset \) P) , then \(xy\in P\) or \(z\in P\). In addition to give many properties and characterizations of these ideals, we also determine rings in which every proper ideal is weakly 1-absorbing prime. We show that weakly 1-absorbing prime ideals are really of interest in decomposable rings.

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  1. Department of Mathematics, Nelson Mandela University, Port Elizabeth, South Africa

    Nico Groenewald

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  1. Nico Groenewald
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Correspondence to Nico Groenewald.

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Communicated by Sergio R. López-Permouth.

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Groenewald, N. 1-absorbing prime ideals and weakly 1-absorbing prime ideals in noncommutative rings. São Paulo J. Math. Sci. 17, 871–887 (2023). https://doi.org/10.1007/s40863-022-00348-2

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