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In mathematics, a derivation ∂ {\displaystyle \partial } of a commutative ring A {\displaystyle A} is called a locally nilpotent derivation if every element of A {\displaystyle A} is annihilated by some power of ∂ {\displaystyle \partial }.
One motivation for the study of locally nilpotent derivations comes from the fact that some of the counterexamples to Hilbert's 14th problem are obtained as the kernels of a derivation on a polynomial ring.
Over a field k {\displaystyle k} of characteristic zero, to give a locally nilpotent derivation on the integral domain A {\displaystyle A} , finitely generated over the field, is equivalent to giving an action of the additive group {\displaystyle } to the affine variety X = Spec {\displaystyle X=\operatorname {Spec} }. Roughly speaking, an affine variety admitting "plenty" of actions of the additive group is considered similar to an affine space.