nil ideal

A nil ideal is an ideal made entirely of elements that vanish when raised to some power — every single member is nilpotent, though possibly to a different exponent. It collects the ring's 'infinitesimal' or 'ghost' elements, ones that are nonzero yet become zero after enough self-multiplication.

An ideal I of a ring R is nil if every element x in I satisfies x^n = 0 for some positive integer n (the n may depend on x). This is weaker than being a nilpotent ideal, which requires a single uniform exponent: I is nilpotent if I^m = 0 for some fixed m, meaning every product of m elements of I vanishes. Every nilpotent ideal is nil, but a nil ideal need not be nilpotent.

Nil ideals sit inside the Jacobson radical: any nil ideal is contained in J(R), because 1 − x is a unit whenever x is nilpotent. In a commutative ring the set of all nilpotents is itself an ideal, the nilradical, which is the largest nil ideal and equals the intersection of all prime ideals. For Noetherian rings the nilradical is nilpotent, so the distinction between nil and nilpotent collapses; in general rings it genuinely persists, and the Köthe conjecture about whether the sum of two nil left ideals is nil remains famously open.