An overnilpotent radical theory for near-rings

Veldsman, Stefan

doi:10.1016/0021-8693(91)90137-w

Cited by 15 publications

(17 citation statements)

References 11 publications

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“…Ideals are denoted by I < N; essential ideals by I <oN ( / i s essential if / n / ^ 0 for all 0 ^ J <N). From [17] we recall that a near-rings N is called quasi semi-equiprime if (0: N) N [6] and Groenewald [5] we need the following definitions.…”

Section: Special Radicalsmentioning

confidence: 99%

“…Consequently any hereditary class of equiprime near-rings which is closed under essential extensions in a special class. Several such examples can be found in [17], and amongst these are the 3-primitive near-rings. Further examples are the class of 2-primitive near-rings and the class of simple near-rings with identity.…”

Section: Special Radicalsmentioning

confidence: 99%

“…From [17, Theorem 2.10] we know that the subdirect closure of a weakly special class is again a weakly special class; hence we have As usual, for a radical class 32 and a near-ring N, 32 (N) denotes the radical of N with respect to 32, that is, 32{N) = £ ( / < N\I € 32). Since any special radical class is an overnilpotent radical, we have from [ 17] 1.6 THEOREM.…”

Section: Special Radicalsmentioning

confidence: 99%

“…However, recent results seem to indicate that the starting point may not be the correct one. In [17], overnilpotent radical classes of near-rings were denned which also generalize the supernilpotent radicals of rings. The starting point here is via weakly special classes which are classes of quasi semi-equiprime near-rings (denned below).…”

mentioning

confidence: 99%

“…Special radicals and matrix near-rings 357 natural examples were given in [17]. In fact, all the known ideal-hereditary radicals are overnilpotent radicals.…”

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confidence: 99%

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Special radicals and matrix near-rings

Veldsman

1992

J Aust Math Soc A

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Special radical classes of near-rings are denned and investigated. It is shown that our approach, which differs from previous ones, does cater for all the well-known radicals of near-rings. Moreover, most of the desirable properties from their ring theory counterpart are retained. The relationship between the special radical of a near-ring and the corresponding matrix near-ring is given. Special radicals in the variety of rings is a specialization of the supernilpotent radicals of rings. The need for supernilpotent radicals arose to discard some pathological radicals. Supernilpotent radicals are hereditary and contain all the nilpotent rings. These two properties (or sometimes just the second) are the denning conditions for a supernilpotent radical in the variety of nearrings. Unfortunately this approach lacks several desirable conclusions, for example, supernilpotent radicals need not have hereditary semisimple classes in the variety of near-rings (cf. [4]). At first it appeared that this is the price to pay for considering supernilpotent radicals in this more general variety. However, recent results seem to indicate that the starting point may not be the correct one. In [17], overnilpotent radical classes of near-rings were denned which also generalize the supernilpotent radicals of rings. The starting point here is via weakly special classes which are classes of quasi semi-equiprime near-rings (denned below). This approach has several advantages; the overnilpotent radicals are always ideal-hereditary. A host of

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Section: Special Radicalsmentioning

confidence: 99%

Section: Special Radicalsmentioning

confidence: 99%

Section: Special Radicalsmentioning

confidence: 99%

mentioning

confidence: 99%

“…Special radicals and matrix near-rings 357 natural examples were given in [17]. In fact, all the known ideal-hereditary radicals are overnilpotent radicals.…”

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confidence: 99%

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