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Let  R  be a prime algebra over a commutative ring  K , let  I  be a nonzero ideal of  R  and letbe a multilinear polynomial over  K  in  k  non-commuting indeterminates which is not central-valued on  R . Suppose thatand  d  is a derivation of  R  such thatfor all, where,  s ,  t ,  m  are fixed positive integers andis the centre of  R . It is shown that  a  = 0 or  d  = 0 except when, thematrix ring over a field  F . This result gives a natural generalization of several well-known theorems in the literature. Moreover, with this we give an affirmative answer to the open conjecture recently raised by Huang in [Derivations with annihilator conditions on Lie ideals in prime rings. J Algebra Appl. 2020;19:2050025].

Derivations with annihilator conditions on multilinear polynomials