Nilpotence is a property expressing that a repeated algebraic operation terminates after finitely many steps. Its precise meaning depends on the kind of object.
For a group, nilpotence means that the group upper central series reaches the whole group; such an object is a nilpotent group. For a Lie algebra, the corresponding condition uses the Lie algebra upper central series and defines a nilpotent Lie algebra. For an algebraic loop, reaching the whole structure in the loop upper central series defines central nilpotence (Semanišinová and Stanovský 2023).
For an element of a ring, nilpotence instead means that some positive integer power is zero. This defines a nilpotent element, and applies in particular to a nilpotent matrix.
