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Some remarks on radicals of rings with chain conditions

Some remarks on radicals of rings with chain conditions Acta Mathematica Academiae Scientiarum Hungaricae Tomus 25 (3--4), (1974), pp. 263--268, SOME REMARKS ON RADICALS OF RINGS WITH CHAIN CONDITIONS By B. J. GARDNER (Hobart) Introduction DIVlNSKY [2] has described those general radicals which have the classical effect on artinian rings. The smallest such radical class is (clearly) the lower radical class N defined by the class of nilpotent rings which are (Jacobson) radicals of artinian rings. Divinsky proved that N properly contains the lower radical class determined by all simple zerorings and is properly contained in the Baer lower radical class N, the latter inequality being demonstrated by means of a moderately complicated example. Using a result from [6] we prove in w that the rings in @ are characterized by their membership of N and a condition on their additive groups. From this it follows that while not hereditary, N has the property that pure ideals of N-rings are N-rings. In w the effect on artinian rings of a fairly large selection of radicals is con- sidered. It is shown that every hereditary radical coincides on artinian rings with the lower radical determined by a class of rings selected from the following: full matrix rings over division http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

Some remarks on radicals of rings with chain conditions

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF01886083
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Abstract

Acta Mathematica Academiae Scientiarum Hungaricae Tomus 25 (3--4), (1974), pp. 263--268, SOME REMARKS ON RADICALS OF RINGS WITH CHAIN CONDITIONS By B. J. GARDNER (Hobart) Introduction DIVlNSKY [2] has described those general radicals which have the classical effect on artinian rings. The smallest such radical class is (clearly) the lower radical class N defined by the class of nilpotent rings which are (Jacobson) radicals of artinian rings. Divinsky proved that N properly contains the lower radical class determined by all simple zerorings and is properly contained in the Baer lower radical class N, the latter inequality being demonstrated by means of a moderately complicated example. Using a result from [6] we prove in w that the rings in @ are characterized by their membership of N and a condition on their additive groups. From this it follows that while not hereditary, N has the property that pure ideals of N-rings are N-rings. In w the effect on artinian rings of a fairly large selection of radicals is con- sidered. It is shown that every hereditary radical coincides on artinian rings with the lower radical determined by a class of rings selected from the following: full matrix rings over division

Journal

Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals

Published: Sep 1, 1974

References

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    Armendariz, E. P.; Leavitt, W. G.
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    Divinsky, N.
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    Divinsky, N. J.
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    Fuchs, L.
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    Gardner, B. J.
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    Gardner, B. J.
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    Jenkins, T. L.; Kreiling, D.
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    Курош, А. Г.
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    Szász, F.
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