Derivations In Prime Γ-rings

Γ-ring is a generalization of ring. In1964, Nobusawa defined Γ-ring, which was ex-tended by Barnes in1966. Since then, many people are interested in Gamma-ring and beginto study the properties of Γ-ring. In recent years, some authors generalize some results inring theory to Γ-ring, and they get many important results.Derivation is an important topic in ring theory. In1957, Posner proved that if a non-trivial derivation d satisfies [a,d(a)]∈Z(R) for all a∈R, then R is commutative. Later,many people generalize Posner’s theorem in diferent directions, however, most of those areon (semi)-prime rings. In2004, Uc kun,O¨ztu¨rk and Jun proved that if a Γ-near-ring R hasno nonzero zero-divisors and has a derivation d of R satisfying [x,d(x)]γ=0for all x∈Randγ∈Γ, then (R,+) is abelian. They also proved that in a prime Γ-near-ring R, if a non-trivial derivation d satisfies d(x)∈Z(R) for all x∈R, then (R,+) is abelian. Furthermore, ifcharR2, then R is commutative. In this paper, we generalize Posner’s theorem to primeΓ-rings. We prove that in a prime Γ-ring R with charR2and Z(R){0}, if a nontrivialderivation d satisfies [a,d(a)]α∈Z(R) for all a∈R, α∈Γ, then R is commutative.In2007, Asci and Ceran proved that if R is a prime Γ-ring with charR2,3and d(I) I,d2(I) Z(R), where I is an ideal of R and d is a left derivation, then R is commutative.Many authors also study maps and polynomial identities on (semi)-prime ring. Let R be aΓ-ring, a, b, c∈R, α, β∈Γ, the commutator formulas are as follows.[a,bαc]β=[a, b]βαc+bα[a, c]β+bβaαc bαaβc,[aαb,c]β=[a, c]βαb+aα[b, c]β+aαcβb aβcαb.In2009, Paul and Halder proved that in a semiprime Γ-ring R, if aαbβc=aβbαc for all a,b,c∈R, α, β∈Γ, then any left derivation of R must map R into its center. In this paper, weremove this assumption and prove that a left derivation of a semiprime Γ-ring R must map Rinto its center. We also prove that if a prime Γ-ring R admits a nonzero left derivation, thenR is commutative.In1992, Bell and Mason defined strong commutativity preserving map on a ring. LetR be a ring, f: Râ†'R be a map and S be a subset of R. If [f (x),f (y)]=[x, y] for allx,y∈S, then f is called a strong commutativity preserving (scp) map on S. In1994, Belland Daif proved that if a derivation d is a scp map on a nonzero right ideal U of a semiprimering R, then U Z(R). They also proved that if an endomorphism T is strong commutativitypreserving on a nonzero ideal I of a semiprime ring R and T is not the identity map onthe ideal I T1(I), then R must contain a nonzero central ideal. In the same year, Bresˇarand Miers also described strong commutativity preserving additive map on semiprime rings.They proved that if an additive map f: Râ†'R is strong commutativity preserving on R,then f (x)=λx+δ(x), where λ∈C, λ2=1, δ: Râ†'C is an additive map, and C is theextended centroid of R. Let R be a Γ-ring, f: Râ†'R be a map and S be a subset of R.If [f (x),f (y)]α=[x, y]αfor all x,y∈S, α∈Γ, then f is called a strong commutativitypreserving (scp) map on S. In this paper, we study derivations that are strong commutativitypreserving on a prime Γ-ring and endomorphisms that are strong commutativity preservingon semiprime Γ-ring. We prove that if a derivation d is strong commutativity preserving on anonzero ideal I of a prime Γ-ring R, and d(I) I, then R is commutative. We also prove thatif there exists a scp derivation on a semiprime Γ-ring R, then R is commutative. Moreover,we prove that an endomorphismδon a semiprime ring R is strong commutativity preservingon R if and only if there is a mapμ: Râ†'Z(R) such thatδ(x)=x+μ(x) for all x∈R.In1969, Herstein defined the Lie structure on a associative ring R by the Lie product[x,y]=xy yx. Let R be an associative ring, A be a Lie subring of R and L be an additivesubgroup of A. If [x,a]=xa ax∈L for all x∈L, a∈A, then L is called a Lie idealof A. In2010, Paul and Uddin defined the Lie structure of a Γ-ring R by a new product[x,y]α=xαy yαx for all x, y∈R, α∈Γ. Let R be a Γ-ring, S be a Lie sub-Γ-ring of R andL be an additive subgroup of S. If [x,s]α=xαs sαx∈L for all x∈L, s∈S, α∈Γ, then L is a Lie ideal of S. They studied Lie ideals of a simple Γ-ring and proved that in a simpleΓ-ring R with charR2, if L is a Lie ideal of R, then either L Z(R) or L [R, R]Γ. In1988,Lanski proved that in a prime ring R, if L is a noncommutative Lie ideal of R and I is an idealof R generated by [L,L], then [I,I] L. In this paper, we study ideals and noncommutativeLie ideals in a Γ-ring. We prove that for every noncommutative Lie ideal L of a Γ-ring R,there exists a nonzero ideal I of R such that L [I,I]Γ.The main results of this paper are as follows.Theorem2.1Let R be a prime Γ-ring, I be a nonzero ideal of R and d be a derivationof R. If d is strong commutativity preserving on I and d(I) I, then R is commutative.Theorem2.4Let R be a prime Γ-ring with charR2, d be a nonzero derivation of R.If [a,d(a)]α∈Z(R) for all a∈R, α∈Γ and Z(R){0}, then R is commutative.Theorem3.1Let R be a semiprime Γ-ring, d be a left derivation of R. Then d(R) Z(R).Theorem3.2Let R be a prime Γ-ring. If there exists a nonzero left derivation on R,then R is commutative.Theorem4.1Let R be a semiprime Γ-ring. If there exists a strong commutativitypreserving derivation on R, then R is commutative.Theorem4.2Let R be a semiprime Γ-ring and δ be an endomorphism of R. Then δ isstrong commutativity preserving on R if and only if there exists a mapμ: Râ†'Z(R) suchthatδ(x)=x+μ(x) for all x∈R.Theorem5.1Let R be a Γ-ring and L be a noncommutative Lie ideal of R. Then thereexists a nonzero ideal I of R such that L [I,I]Γ.