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Homenil and nilpotent ideals

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# nil and nilpotent ideals

A ring $R$ is *nil* if every element in $R$ is nilpotent. Similarly, a one- or two-sided ideal is called *nil* if each of its elements is nilpotent.

A ring $R$ [resp. a one- or two sided ideal $A$] is *nilpotent* if $R^{n}=0$ [resp. $A^{n}=0$] for some positive integer $n$.

A ring or an ideal is *locally nilpotent* if every finitely generated subring is nilpotent.

The following implications hold for rings (or ideals):

$\text{nilpotent}\quad\Rightarrow\text{locally nilpotent}\quad\Rightarrow\text{nil}$ |

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 nilpo

Related:

KoetheConjecture

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

16N40*no label found*

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new correction: Define Galois correspondence by porton

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new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

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new correction: Many corrections by Smarandache

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new question: how to contest an entry? by zorba

new question: simple question by parag

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