### Abstract

The usual definition of primeness for near-rings does not lead to a Kurosh-Amitsur radical class for zerosymmetric near-rings (cf. Kaarli and Kriis[5]). In this paper, we define equiprime near-rings, which are another generalization of primeness in rings. Various results are proved, amongst others, for zerosymmetric near-rings: If N is a near-ring and A ◃ B ◃ N such that B/A is equiprime, then A ◃ N. We define [formula omitted]. Then P_{*} is a Kurosh-Amitsur radical for which [formula omitted] holds for all ideals I of N. Moreover, [formula omitted], where I_{3}, is the Jacobson-type radical which was introduced by Holcombe, with equality if N has the descending chain condition on N-sub-groups.

Original language | English |
---|---|

Pages (from-to) | 3111-3122 |

Number of pages | 12 |

Journal | Communications in Algebra |

Volume | 18 |

Issue number | 9 |

DOIs | |

Publication status | Published - Jan 1 1990 |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Communications in Algebra*,

*18*(9), 3111-3122. https://doi.org/10.1080/00927879008824063

**A kurosh-amitsur prime radical for near-rings.** / Booth, G. L.; Groenewald, N. J.; Veldsman, S.

Research output: Contribution to journal › Article

*Communications in Algebra*, vol. 18, no. 9, pp. 3111-3122. https://doi.org/10.1080/00927879008824063

}

TY - JOUR

T1 - A kurosh-amitsur prime radical for near-rings

AU - Booth, G. L.

AU - Groenewald, N. J.

AU - Veldsman, S.

PY - 1990/1/1

Y1 - 1990/1/1

N2 - The usual definition of primeness for near-rings does not lead to a Kurosh-Amitsur radical class for zerosymmetric near-rings (cf. Kaarli and Kriis[5]). In this paper, we define equiprime near-rings, which are another generalization of primeness in rings. Various results are proved, amongst others, for zerosymmetric near-rings: If N is a near-ring and A ◃ B ◃ N such that B/A is equiprime, then A ◃ N. We define [formula omitted]. Then P* is a Kurosh-Amitsur radical for which [formula omitted] holds for all ideals I of N. Moreover, [formula omitted], where I3, is the Jacobson-type radical which was introduced by Holcombe, with equality if N has the descending chain condition on N-sub-groups.

AB - The usual definition of primeness for near-rings does not lead to a Kurosh-Amitsur radical class for zerosymmetric near-rings (cf. Kaarli and Kriis[5]). In this paper, we define equiprime near-rings, which are another generalization of primeness in rings. Various results are proved, amongst others, for zerosymmetric near-rings: If N is a near-ring and A ◃ B ◃ N such that B/A is equiprime, then A ◃ N. We define [formula omitted]. Then P* is a Kurosh-Amitsur radical for which [formula omitted] holds for all ideals I of N. Moreover, [formula omitted], where I3, is the Jacobson-type radical which was introduced by Holcombe, with equality if N has the descending chain condition on N-sub-groups.

UR - http://www.scopus.com/inward/record.url?scp=0012100444&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0012100444&partnerID=8YFLogxK

U2 - 10.1080/00927879008824063

DO - 10.1080/00927879008824063

M3 - Article

AN - SCOPUS:0012100444

VL - 18

SP - 3111

EP - 3122

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 9

ER -