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):

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