nil and nilpotent ideals

An element $x$ of a ring is nilpotent if $x^n=0$ for some positive integer $n$.

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):

$$\text{nilpotent}\mathit{\hspace{1em}}\Rightarrow \text{locally nilpotent}\mathit{\hspace{1em}}\Rightarrow \text{nil}$$

Title	nil and nilpotent ideals
Canonical name	NilAndNilpotentIdeals
Date of creation	2013-03-22 13:13:25
Last modified on	2013-03-22 13:13:25
Owner	mclase (549)
Last modified by	mclase (549)
Numerical id	6
Author	mclase (549)
Entry type	Definition
Classification	msc 16N40
Related topic	KoetheConjecture
Defines	nil
Defines	nil ring
Defines	nil ideal
Defines	nil right ideal
Defines	nil left ideal
Defines	nil subring
Defines	nilpotent
Defines	nilpotent element
Defines	nilpotent ring
Defines	nilpotent ideal
Defines	nilpotent right ideal
Defines	nilpotent left ideal
Defines	nilpotent subring
Defines	locally nilpotent
Defines	locally nilpotent ring
Defines	locally nilpo