Skew N-Derivations On Semiprime Rings

The notion of symmetric biderivation was introduced by Maksa in1980. Let R be a ring. A biadditive mapping D(.,.):R×R�'R is called a symmetric biderivation if the mapping x (?)D(x, y) is a derivation for an arbitrary fixed y∈R.Vukman generalized the classical Posner Theorem to symmetric biderivations in prime and semiprime rings. Thereafter many literatures focused on biderivations of prime and semiprime rings. Bresar gave the most important results of the structure of biderivations on semiprime rings.Bresar Theorem Let R be a semiprime ring and let B:R x R�'R be a biderivation. Then there exist an idempotent ε∈C and an element μ∈C such that the algebra (1-ε)R is commutative and εB(x,y)=με[x,y] for all x, y∈R. Particularly, if R is a noncommutative prime ring, then there exists λ∈C such that B(x, y)=λ[x, y] for all x, y∈R.In2007, Jung and Park considered permuting3-derivations on prime and semiprime rings and obtained the following results.Theorem A (Jung and Park) Let R be a noncommutative3-torsion free semiprime ring and let I be a nonzero two-sided ideal of R. Suppose that there exists a permuting3-derivation△:R×R×R�'R such that8is centralizing on I([δ(x), x]∈Z(R), x∈I), where8is the trace of△. Then δ is commuting on I([δ(x), x]=0, x∈I).Theorem B (Jung and Park) Let R be a3!-torsion free prime ring and let I be a nonzero two-sided ideal of R. Suppose that there exists a nonzero permuting3-derivation△:R×R×R�'R such that8is centralizing on I, where δ is the trace of△. Then R is commutative.Park obtained the similar results for permuting4-derivations on prime and semiprime rings. In2009, Park considered permuting n-derivations on prime and semiprime rings.This paper depends on Bresar’s results. In view of his proofs, we give some modifica-tions of his theorems in order to apply them better. Concretely we get two theorems. Theorem2.1Let S be a set and R be a semiprime ring. If functions f and g of S into R satisfy that f(s)xg(t)=ξg(s)xf(t) for all s,t∈S, x∈R, where ξ∈C is an invertible element, then there exist idempotents ε1,ε2,ε3∈C and an invertible element λ∈C such that εjεj=0, for i≠j, ε1+ε2+ε3=1, and ε1/(s)=λε1g(s),ε2g(s)=0, ε3f(5)=0,(1-ξ)ε1f(s)=0holds for all s∈S.Theorem2.2Let R be a semiprime ring with an automorphism σ, and let B:R×R�'R be a σ-biderivation. Then there exist idempotents,ε1,ε2,ε3∈C and invertible elements P∈Qmr, κ∈C such that●ε1+ε2+ε3=1,ε1ε2=ε1ε3=ε2ε3=0,●ε1B(x,y)=ε1p[x,y], ε2B(x,y)=0, andε3[x,y]=0for all x, y∈R.In this paper, we consider the skew n-derivations on prime and semiprime rings, which are the generalization of the biderivations on prime and semiprime rings. We obtain main theorem.Theorem3.1A skew n-derivation (n≥3) on a semiprime ring R must map into the center of R.As a corollary, we prove that an arbitrary skew n-derivation (n≥3) on a noncommuta-tive prime ring R must be zero. These results can reveal the reason why Theorem A, B and the results in the literatures hold.Theorem3.2A prime ring with a nonzero skew n-derivation (n≥3) must be commu-tative.