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nil and nilpotent ideals
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(Definition)
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An element  of a ring is nilpotent if  for some positive integer  .
A ring  is nil if every element in  is nilpotent. Similarly, a one- or two-sided ideal is called nil if each of its elements is nilpotent.
A ring  [resp. a one- or two sided ideal  ] is nilpotent if  [resp.  ] for some positive integer  .
A ring or an ideal is locally nilpotent if every finitely generated subring is nilpotent.
The following implications hold for rings (or ideals):
nilpotent  locally nilpotent  nil
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"nil and nilpotent ideals" is owned by mclase.
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(view preamble)
See Also: Koethe conjecture
| Also defines: |
nil, nil ring, nil ideal, nil right ideal, nil left ideal, nil subring, nilpotent, nilpotent element, nilpotent ring, nilpotent ideal, nilpotent right ideal, nilpotent left ideal, nilpotent subring, locally nilpotent, locally nilpotent ring, locally nilpotent ideal, locally nilpotent right ideal, locally nilpotent left ideal, locally nilpotent subring |
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Cross-references: implications, subring, finitely generated, ideal, two-sided ideal, integer, positive, ring
There are 20 references to this entry.
This is version 3 of nil and nilpotent ideals, born on 2002-12-08, modified 2003-09-04.
Object id is 3690, canonical name is NilAndNilpotentIdeals.
Accessed 20194 times total.
Classification:
| AMS MSC: | 16N40 (Associative rings and algebras :: Radicals and radical properties of rings :: Nil and nilpotent radicals, sets, ideals, rings) |
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Pending Errata and Addenda
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