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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 question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

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new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

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new image: plot W(t) = P(waiting time <= t) by robert_dodier

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier