In the mathematical field of commutative algebra, an ideal I in a commutative ring A is locally nilpotent at a prime ideal p if the ideal is nilpotent in a Zariski open neighborhood of p. That is, I is nilpotent at p if Ip, the localization of I at p, is a nilpotent ideal in Ap.
In non-commutative algebra and group theory, an algebra or group is locally nilpotent if and only if every finitely generated subalgebra or subgroup is nilpotent. The subgroup generated by the normal locally nilpotent subgroups is called the Kurt Hirsch-Plotkin radical and is the generalization of the Fitting subgroup to groups without the ascending chain condition on normal subgroups. In non-commutative ring theory, a locally nilpotent ideal is called a nil ideal.