Nil ideal

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

Nil ideals measure the “nilpotent part” of a ring and are always contained in the nilradical in the commutative case. They appear naturally as radicals of Artinian local rings and as Jacobson radical components in finite-dimensional algebras.

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.