On some equations related to derivations in rings

نویسندگان

  • Joso Vukman
  • Irena Kosi-Ulbl
چکیده

Throughout this paper, R will represent an associative ring with center Z(R). A ring R is n-torsion free, where n > 1 is an integer, in case nx = 0, x ∈ R implies x = 0. As usual the commutator xy− yx will be denoted by [x, y]. We will use basic commutator identities [xy,z] = [x,z]y + x[y,z] and [x, yz] = [x, y]z+ y[x,z]. Recall that a ring R is prime if aRb = (0) implies that either a = 0 or b = 0, and is semiprime if aRa = (0) implies a = 0. An additive mapping D : R→ R is called a derivation if D(xy) = D(x)y + xD(y) for all pairs x, y ∈ R, and is called a Jordan derivation in case D(x2) =D(x)x+ xD(x) for all x ∈ R. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein [11, Theorem 3.1] asserts that any Jordan derivation on a 2-torsion free prime ring is a derivation (see [7] for an alternative proof). Cusack [9, Corollary 5] has generalized Herstein’s theorem to 2-torsion free semiprime rings (see [4] for an alternative proof). A mapping F of a ring R into itself is called commuting (centralizing) on R in case [F(x),x] = 0 ([F(x),x] ∈ Z(R)) holds for all x ∈ R. The theory of commuting and centralizing mappings was initiated by a result of Posner [12, Theorem 2] (Posner’s second theorem), which states that the existence of a nonzero centralizing derivation D : R→ R, where R is a prime ring, forces the ring to be commutative. Vukman has proved the following result.

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عنوان ژورنال:
  • Int. J. Math. Mathematical Sciences

دوره 2005  شماره 

صفحات  -

تاریخ انتشار 2005