Nil ideal

In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if all of its elements is nilpotent, i.e for each ${\displaystyle a\in I}$ exists natural number n for which ${\displaystyle a^{n}=0.}$ If all elements of a ring is nilpotent (this is possible only for rings without a unit), then the ring is called a nil ring.

The nilradical of a commutative ring is an example of a nil ideal; in fact, it is the ideal of the ring maximal with respect to the property of being nil. Unfortunately the set of nilpotent elements does not always form an ideal for noncommutative rings. Nil ideals are still associated with interesting open questions, especially the unsolved Köthe conjecture.