nilpotent ideal

Some ideals, if you multiply them by themselves enough times, collapse entirely to zero. They are the algebraic analogue of a substance that vanishes after a fixed number of self-reactions. A nilpotent ideal is exactly such an ideal: a finite power of it is the zero ideal. These ideals are pure obstruction — they are the gunk a ring must shed before it can be semisimple.

Precisely, an ideal I of a ring R is nilpotent if I^n = 0 for some positive integer n, where I^n is the ideal generated by all products of n elements of I. Note this is stronger than asking each element of I to be nilpotent (that weaker condition defines a nil ideal): nilpotency demands a single uniform exponent that annihilates all products at once.

Every nilpotent ideal lies inside the Jacobson radical, and in an Artinian ring the radical is itself nilpotent, so for finite-length rings the radical is the unique largest nilpotent ideal. This is the precise sense in which a semisimple Artinian ring is one with no nilpotent ideals: killing the radical and getting zero is exactly the condition that no self-shrinking gunk remains.

An honest distinction: nilpotent implies nil, but not conversely in general. For a left Artinian ring the two notions coincide for ideals, which is why finite-dimensional algebra is comfortable; but Köthe's conjecture, asking whether the sum of two nil left ideals is always nil, remains open in full generality, a reminder that nil and nilpotent diverge in the wild.