Characterization Of Locally Nilpotent Derivations Over Algebras

In this paper, we investigate into the locally nilpotent derivations of algebras over the field F with char F=0. Firstly, we give the characterization (sd )2 + (cd)2 = I of linear translations sd and cd over general algebras (where I is identity map) in the first part, that is, the characterization of locally nilpotent derivation is shown. In the second part, we study locally nilpotent derivations of prime algebras and prove that, if the prime algebra A has the unit, the A has no zero divisors if and only if each nilpotent derivation of A is 0. After that, locally nilpotent derivations are investigated and several sufficient conditions becoming nilpotent derivations are given, (1) Suppose that d is a derivation of finite-dimensionally prime algebra A. If a ∈ 0 satisfies d(a) =0 and each x ∈ A n(x) > 1 such that adn (x)=0, then d is nilpotent. (2) Suppose that d1, d2 are derivations of the finite-dimensionally prime algebra A, d1∈0 and d\d=dd1. If x∈A, 3n(x)>l, such that d,d2n(x)(x)=0, then d is a nilpotent one. (3) Suppose d1, d2 are ones of the finite- dimensionally prime algebra A, and d1d2=d2d1. If V x ∈ A, n(x), m(x) > 1, such that of1n (x )d2m(x)=0, then one of d1 and d2 is nilpotent at least. Finally, we investigate the structure of locally nilpotent derivation of the polynomial algebra F[x]; and the formula of each locally nilpotent one is a0 (where c0∈F , d is the derivative of F[x]).