Abstract: Let ${\cal N}$ and ${\cal M}$ be two nests on real or complex Banach spaces X and Y, respectively, and $\Phi$ be an additive map between ideals Alg$_{\cal F}{\cal N}$ and Alg$_{\cal F}{\cal M}$ of finite rank operators in nest algebras Alg${\cal N}$ and Alg${\cal M}$, of which the range contains all
rank-1 nilpotent operators in Alg${\cal M}$. The authors show that if $\Phi$ is rank-1 nilpotency preserving in both directions, then $\Phi$ has the form either $\Phi(x\otimes f)=Ax\otimes Cf$ for every rank-1 nilpotent operator $x\otimes {\rm Alg}_{\cal F}{\cal N}$ or $\Phi(x\otimes f)=Af\otimes Cx$ for every rank-1 nilpotent operator $x\otimes f\in {\rm Alg}_{\cal F}{\cal N}$,
where $A$ and $C$ are certain $\tau$-linear operators with an automorphism $\tau$ of the underlying field. And the authors obtain particularly a characterization of such $\Phi$ if it is continuous, $X$ and $Y$ are Hilbert spaces with dimX≥ 6.

Key words: Additive map, Rank one nilpotent operator, Nest algebra