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| An element $x$ of a ring is \emph{nilpotent} if $x^n = 0$ for some positive integer $n$. |
An element $x$ of a ring is \emph{nilpotent} if $x^n = 0$ for some positive integer $n$. |
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| A ring $R$ is \emph{nil} if every element in $R$ is nilpotent. Similarly, a one- or two-sided ideal is called \emph{nil} if each of its elements is nilpotent. |
A ring $R$ is \emph{nil} if every element in $R$ is nilpotent. Similarly, a one- or two-sided ideal is called \emph{nil} if each of its elements is nilpotent. |
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A ring $R$ [resp. a one- or two sided ideal $S$] is \emph{nilpotent} if $R^n = 0$ [resp. $A^n = 0$] for some positive integer $n$. |
| A ring $R$ [resp. a one- or two sided ideal $A$] is \emph{nilpotent} if $R^n = 0$ [resp. $A^n = 0$] for some positive integer $n$. |
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| A ring or an ideal is \emph{locally nilpotent} if every finitely generated subring is nilpotent. |
A ring or an ideal is \emph{locally nilpotent} if every finitely generated subring is nilpotent. |
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| The following implications hold for rings (or ideals): |
The following implications hold for rings (or ideals): |
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| $$\text{nilpotent} \quad\Rightarrow \text{locally nilpotent} \quad\Rightarrow \text{nil}$$ |
$$\text{nilpotent} \quad\Rightarrow \text{locally nilpotent} \quad\Rightarrow \text{nil}$$ |
