Weakly Prime and Weakly Completely Prime Ideals of Noncommutative Rings

Groenewald, N. J.

doi:10.24330/ieja.768127

Cited by 7 publications

(3 citation statements)

References 4 publications

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“…Exploring the extension of mathematical concepts to noncommutative rings has garnered signifcant attention from researchers. Notably, works such as [3,4] have delved into the generalization of weakly prime ideals in noncommutative rings. In the context of these generalizations, an ideal P of a ring R is considered weakly prime if, for any ideals J and K of R, the condition 0 ≠ J. K ⊆ P implies either J ⊆ P or K ⊆ P, as demonstrated in [4].…”

Section: Introductionmentioning

confidence: 99%

A Note on Weakly Semiprime Ideals and Their Relationship to Prime Radical in Noncommutative Rings

Abouhalaka

2024

Journal of Mathematics

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In this paper, we introduce the concept of weakly semiprime ideals and weakly n-systems in noncommutative rings. We establish the equivalence between an ideal P being a weakly semiprime ideal and R−P being a weakly n-system. We provide alternative definitions and explore the properties of weakly semiprime ideals. Additionally, we investigate scenarios where all ideals in a given ring are weakly semiprime and demonstrate that in Noetherian rings, where every ideal is weakly semiprime, the prime radical and the Jacobson radical coincide.

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Section: Introductionmentioning

confidence: 99%

A Note on Weakly Semiprime Ideals and Their Relationship to Prime Radical in Noncommutative Rings

Abouhalaka

2024

Journal of Mathematics

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“…Massouros and Yaqoob [19] presented the study of algebraic structures, left/right almost groups, and hypergroups equipped with the inverted associativity axiom, and they analyzed the algebraic properties of these special nonassociative hyperstructures. We refer readers to see if they want to learn more about LA-rings [9,[20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

Some Developments in the Field of Homological Algebra by Defining New Class of Modules over Nonassociative Rings

Razzaque

Rehman

2022

Journal of Mathematics

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The LA-module is a nonassociative structure that extends modules over a nonassociative ring known as left almost rings (LA-rings). Because of peculiar characteristics of LA-ring and its inception into noncommutative and nonassociative theory, drew the attention of many researchers over the last decade. In this study, the ideas of projective and injective LA-modules, LA-vector space, as well as examples and findings, are discussed. We construct a nontrivial example in which it is proved that if the LA-module is not free, then it cannot be a projective LA-module. We also construct free LA-modules, create a split sequence in LA-modules, and show several outcomes that are connected to them. We have proved the projective basis theorem for LA-modules. Also, split sequences in projective and injective LA-modules are discussed with the help of various propositions and theorems.

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“…Most recently in 2020, Groenewald [4] gave more properties of weakly prime ideals in noncommutative rings, and introduced the notion of the weakly prime radical of an ideal. Theorem 1.14 states that for an ideal P of any ring R, the following statements are equivalent.…”

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confidence: 99%