# 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):

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