On Class Pairs and Radicals

Birkenmeier, Gary F.; Groenewald, N. J.; Olivier, W. A.

doi:10.1081/agb-100106821

Cited by 3 publications

(3 citation statements)

References 21 publications

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“…Part of this work is an extension to near-rings of the work done by Birkenmeier, Groenewald and Olivier [3].…”

Section: Class Pairsmentioning

confidence: 96%

“…The class pair constructed from M 1 and M 2 is the set (M 1 : M 2 ) = {R : I R and R/I ∈ M 2 imply R/I ∈ M 1 } (see [3]). …”

Section: Class Pairsmentioning

confidence: 99%

“…We will denote the classes of 0-prime, 1-prime, 2-prime, 3-prime and equiprime near-rings by P 0 , P 1 , P 2 , P 3 and P e , respectively. The class of completely prime near-rings is denoted by P c .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Jacobson Near-rings and Special Radicals

2007

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In this paper, we construct special radicals using class pairs of near-rings. We establish necessary conditions for a class pair to be a special radical class. We then define Jacobson-type near-rings and show that in most cases the class of all near-rings of this type is a special radical class. Subsequently, we investigate the relationship between each Jacobson-type near-ring and the corresponding matrix near-ring.

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“…Part of this work is an extension to near-rings of the work done by Birkenmeier, Groenewald and Olivier [3].…”

Section: Class Pairsmentioning

confidence: 96%

“…The class pair constructed from M 1 and M 2 is the set (M 1 : M 2 ) = {R : I R and R/I ∈ M 2 imply R/I ∈ M 1 } (see [3]). …”

Section: Class Pairsmentioning

confidence: 99%

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On Jacobson Near-rings and Special Radicals

2007

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On the prime radicals of near-rings and near-ring modules

Groenewald¹

2016

Nearrings, Nearfields and Related Topics

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Near-Ring Radicals and Class Pairs

Godloza¹,

Groenewald

Olivier

2005

Algebra Colloq.

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For near-ring ideal mappings ρ1 and ρ2, we investigate radical theoretical properties of and the relationship among the class pairs (ρ1: ρ2), [Formula: see text] and (ℛρ2: ℛρ1). Conditions on ρ1 and ρ2 are given for a general class pair to form a radical class of various types. These types include the Plotkin and KA-radical varieties. A number of examples are shown to motivate the suitability of the theory of Hoehnke-radicals over KA-radicals when radical pairs of near-rings are studied. In particular, it is shown that [Formula: see text] forms a KA-radical class, where [Formula: see text] denotes the class of completely prime near-rings and [Formula: see text] the class of 3-prime near-rings. This gives another near-ring generalization of the 2-primal ring concept. The theory of radical pairs are also used to show that in general the class of 3-semiprime near-rings is not the semisimple class of the 3-prime radical.

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On Class Pairs and Radicals

Cited by 3 publications

References 21 publications

On Jacobson Near-rings and Special Radicals

On Jacobson Near-rings and Special Radicals

On the prime radicals of near-rings and near-ring modules

Near-Ring Radicals and Class Pairs

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