Products of derivations which act as Lie derivations on commutators of right ideals
نویسنده
چکیده
Throughout this note, R will be always a prime ring of characteristic different from 2 with center Z(R), extended centroid C, and two-sided Martindale quotient ring Q. Let f : R→ R be additive mapping of R into itself. It is said to be a derivation of R if f (xy)= f (x)y + x f (y), for all x, y ∈ R. Let S⊆ R be any subset of R. If for any x, y ∈ S, f ([x, y])= [ f (x), y] + [x, f (y)], then the mapping f is called a Lie derivation on S. Obviously any derivation of R is a Lie derivation on any arbitrary subset S of R. A typical example of a Lie derivation is an additive mapping which is the sum of a derivation and a central map sending commutators to zero. The well-known Posner first theorem states that if δ and d are two nonzero derivations of R, then the composition (dδ) cannot be a nonzero derivation of R [12, Theorem 1]. An analog of Posner’s result for Lie derivations was proved by Lanski [8]. More precisely, Lanski showed that if δ and d are two nonzero derivations of R and L is a Lie ideal of R, then (dδ) cannot be a Lie derivation of L into R unless char(R)= 2 and either R satisfies s4(x1, . . . ,x4), the standard identity of degree 4, or d = αδ, for α∈ C. This note is motived by the previous cited results. Our main theorem gives a generalization of Lanski’s result to the case when (dδ) is a Lie derivation of the subset [I ,I] into R, where I is a nonzero right ideal of R and the characteristic of R is different from 2. The statement of our result is the following.
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ورودعنوان ژورنال:
- Int. J. Math. Mathematical Sciences
دوره 2006 شماره
صفحات -
تاریخ انتشار 2006