## You are here

Homenil and nilpotent ideals

## Primary tabs

# 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*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Apr 22

new question: Prove that for any sets A, B, and C, An(BUC)=(AnB)U(AnC) by St_Louis

Apr 20

new image: information-theoretic-distributed-measurement-dds.png by rspuzio

new image: information-theoretic-distributed-measurement-4.2 by rspuzio

new image: information-theoretic-distributed-measurement-4.1 by rspuzio

new image: information-theoretic-distributed-measurement-3.2 by rspuzio

new image: information-theoretic-distributed-measurement-3.1 by rspuzio

new image: information-theoretic-distributed-measurement-2.1 by rspuzio

Apr 19

new collection: On the Information-Theoretic Structure of Distributed Measurements by rspuzio

Apr 15

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 question: Prove that for any sets A, B, and C, An(BUC)=(AnB)U(AnC) by St_Louis

Apr 20

new image: information-theoretic-distributed-measurement-dds.png by rspuzio

new image: information-theoretic-distributed-measurement-4.2 by rspuzio

new image: information-theoretic-distributed-measurement-4.1 by rspuzio

new image: information-theoretic-distributed-measurement-3.2 by rspuzio

new image: information-theoretic-distributed-measurement-3.1 by rspuzio

new image: information-theoretic-distributed-measurement-2.1 by rspuzio

Apr 19

new collection: On the Information-Theoretic Structure of Distributed Measurements by rspuzio

Apr 15

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