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  • Author or Editor: G. Booth x
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Abstract

In a previous work [3], the current author, together with P. R. Hall, studied various kinds of primeness in near-rings and sandwich near-rings of continuous functions. In this paper, we study various prime radicals for sandwich near-rings. In certain cases, complete characterisations of the prime, 3-prime, equiprime and strongly equiprime radicals are obtained.

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The Brown-McCoy radical is known to be an ideal-hereditary Kurosh-Amitsur radical in the variety of zerosymmetric near-rings. We define the Brown-McCoy and simplical radicals, and , respectively, for zerosymmetric Γ-near-rings. Both and are ideal-hereditary Kurosh-Amitsur radicals in that variety. IfM is a zerosymmetric Γ-near-ring with left operator near-ringL, it is shown that , with equality ifM has a strong left unity. is extended to the variety of arbitrary near-rings, and and are extended to the variety of arbitrary Γ-near-rings, in a way that they remain Kurosh-Amitsur radicals. IfN is a near-ring andAN, then , with equality ifA if left invariant.

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Authors: G. Booth and N. Groenewald

Abstract  

We define a prime ΓM-module for a Γ-ringM. It is shown that a subsetP ofM is a prime ideal ofM if and only ifP is the annihilator of some prime ΓM-moduleG. s-prime ideals ofM were defined by the first author. We defines-modules ofM, analogous to a concept defined by De Wet for rings. It is shown that a subsetQ ofM is ans-prime ideal ofM if and only ifQ is the annihilator of somes-moduleG ofM. Relationships between prime ΓM-modules and primeR-modules are established, whereR is the right operator ring ofM. Similar results are obtained fors-modules.

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Abstract  

A radical α in the universal class of associative rings is called matric-extensible if α (R n) = (α (R))n for any ring R, and natural number n, where R n denotes the nxn matrix ring with entries from R. We investigate matric-extensibility of the lower radical determined by a simple ring S. This enables us to find necessary and sufficient conditions for the lower radical determined by S to be an atom in the lattice of hereditary matric-extensible radicals. We also show that this lattice has atoms which are not of this form. We then describe all atoms of the lattice, and show that it is atomic.

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It was previously shown that every special radical classR of rings induces a special radical class ?R of G-rings. Amongst the special radical classes of near-rings, there are some, called the s-special radical classes, which induce, special radical classes of G-near-rings by the same procedure as used in the ring case. The s-special radicals of near-rings possess very strong hereditary properties. In particular, this leads to some new results for the equiprime andI 3 radicals.

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Summary  

We continue our study of the lattice of matric-extensible radicals of associative rings. We  find some atoms generated by simple rings of the lattices of all matric-extensible radicals, matric-extensible supernilpotent radicals and matric-extensible special radicals. We consider *-rings, which were previously defined by the second author, and consider when they generate atoms of these lattices.

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Authors: G. L. Booth and N. J. Groenewald
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Abstract  

A class K of rings has the GADS property (i.e., generalized ADS property) if wheneverX& I& R with XK, then there exists B & R with BK such that XBI. Radicals whose semisimple classes have the GADS property are called g-radicals. In this paper, we fully characterize the class of g -radicals. We show that ? is a g-radical if and only if either ? ⊆ I or S? ⊆ I , whereI denotes the class of idempotent rings and S? denotes the semisimple class of ?. It is also shown that the (hereditary) g-radicals form an (atomic) sublattice of the lattice of all radicals.

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