Nil ideal - Knowlpedia

A nil ideal in a ring $R$ is an ideal $I\subseteq R$ such that every element of $I$ is a nilpotent element.

Remarks

In a commutative ring, every nil ideal is contained in the nilradical. A nil ideal need not be nilpotent unless additional hypotheses are imposed.

Examples

In $k[x]/(x^n)$ , the ideal generated by the class of $x$ is a nil ideal.
In the ring of upper triangular $n\times n$ matrices over a field, strictly upper triangular matrices form a nil ideal.
The zero ideal $\{0\}$ is nil in every ring.