Derivations On Triangular Algebras

In2001, Cheung defined triangular algebra. Let R be a commutative ring with identity, A, B be unital algebras over R and M be a nonzero unital (A,B)-bimodule. Then we define to be an associative algebra under matrix-like addition and matrix-like multiplication. An algebra Τ is called a triangular algebra if there exit R-algebras A, B and nonzero (A, B)-bimodule M such that Τ is isomorphic to (OABM). Usually, we denote a triangular algebra by Τ=(OABM)Xie and Cao described the derivations on triangular algebras. Let D, G be derivations on T, then we know that there exist derivations dA,gA on algebra A, dB, gB on algebra B, additive mappings f, h on the nonzero module M and u,v∈M such that for each and for all a∈A, b∈B, x∈M we have In this paper, we always have D and G be derivations on Τ, which have the structures above.In1992, Daif and Bell proved that if a derivation d of a semiprime ring R satisfies d[x, y])=±[x, y] on an ideal I of R, then I is the central ideal of R. In particular, if I=R, then R is commutative. In2003, Quadri considered similar problems on prime: If a prime ring R admits a generalized derivation F which satisfies one of the following generalized identities:(1) F([x,y])=[x,y];(2) F([x,y])=—[x,y];(3) F(xoy)=(xoy);(4) F(x o y)=—(x o y), then R is commutative. In this paper, working on a triangular algebra, we describe the structure of the derivation D, which satisfies D([X, Y])=[X, Y] or D([X,Y])=[D(X),Y}.Vukman generalized the Posner’s theorem to more general situation:(1) Let R be a prime with char R≠2, d be a nonzero derivation of R such that [[d(x),x],x]=0, x E R, then R is commutative;(2) Let R be a prime with char R≠2,3, d be a nonzero derivation of R such that [[d(x),x],x] G Z(R), x∈R, then R is commutative. Huang Yun-Bao simplified Vukman’s conditon on char R≠3, proved that if R be a prime with char R≠2, d be a nonzero derivation of R such that [[d(x),x],x] G Z(R), x∈R, then R is commutative. In1996, Wang Xue-Kuan showed that in a prime R with char R≠2, if nonzero derivations d and g satisfy [d(x),g(y)] G Z(R), x, y E R, then R is commutative. In1998, Hvala proved that in a prime R with char R≠2, if nonzero generalized derivations d and g satisfy [d(x),g(x)]=0, x∈R, then there exists λ∈C (C be extended centroid of R) such that d(x)=\g{x). In this paper we prove the identity [D(X),G(X)]=0and [D(X),G(X)] E Z(Τ) are equivalent, and give some other equivalent conditions. We also describe the structure of the derivations D, G of T which satisfing [[D(X),G(X)],X]=0, and prove that under certain conditions,[[D(X),G(X)],X]=0and [D(X),G(X)] E Z(T) are equivalent. In1997, Lanski proved that if L is a nonzero ideal of semiprime ring R, d is a nonzero derivation of R such that [[[d(xt0), xt1],···], xtn]=0, x E L, t0, t1,···, tn are positive integers, then the ideal of R generated by d(L) and d(R)L is in the cen-ter of R, or d(L)=0. In2004, Xu Xiao-Wei proved that if R is a prime with char R≠2, U1, U2,···, Un be Lie ideals of R, d1, d2,···, dn be derivations of R such that [[···[d1(U1), d2(U2)]),···], dn(Un)](?) Z (Z be center of R), then there exits i E {1,2,···,n} such that Ui C Z. In this paper, we describe the structure of the deriva-tion D of Τ, satisfing either of the generalized Engel condtion [[···[[D(Xm),Xn1],Xn2],·],Xnk]=O and [[···[[D(Xm)Xn-XpG(Xq),Xn1],Xn2],···],Xnk]=0where m, n, p, q, n1, n2,··· nk are any positive inters.In1992, Mason and Bell defined strong commutativity preserving map on a ring. Let/be a map of ring R, S be a subset of R. If [f(x), f(y)]=[x,y] for all x, y E S, then/is called a strong commutativity preserving map on S. Let T=Tri(A, M, B) be a triangular ring, under some mild assumption, that every surjective strong commutativity preserving map Φ: Tâ†'T is of the form Φ(X)=λX+μ(X), where A is in Z(Τ), the center of Τ, A2=1Τ and μ is a map from Τ into Z(Τ). In this paper, we prove that derivation on triangular algebra Τ will not be strong commutativity preserving. We also describe the structure of the derivation D of Τ,which satisfies [D(X), D(Y)]=0.The main results of this paper are the following.Theorem2.1Let T=(OABM) be a triangular algebra which consists of A, M, B, and D bea derivation of T, satisfies the generalized identity D([X,Y])=[X,Y] if and only if f(x)=x for all x E M, and A, B be commutative.Theorem2.2Let T=(OABM) be a triangular algebra which consists of A, M, B, and D be a derivation of Τ, satisfies the generalized identity D([X, Y])=[D(X),Y], then D=0.Theorem2.3Let T=(OABM) be a triangular algebra which consists of A, M, B, and D, G be derivations of T, then the following conditions are equivalent: (i)[D(X),G(X)]∈Z(Τ);(ii)[D(X),G(X)]=O;(iii) For all a∈A, b∈B,m∈M, we have [dA(a),gA(a)]=0,[dB(b),gB(b) O,ugB(b)=vdB(b), dA(a)v=gA(a)u, f(m)gB(b)=h(m)dB(b), dA(a)h(m) gA(a)f(m).Theorem2.4Let T=(OABM) be a triangular algebra which consists of A, M, B and D, G be derivations of Τ. If M is2-torsion free, char A≠2, char B≠2, then the following conditions are equivalent:(i)[[D(X),G(X)],X]=O(ii)[D(X),G(X)],∈Z(Τ)(iii)[D(I),G(X)]=O;(iv) For all a∈A, b∈B, m∈M, we have [dA(a),gA(a)]=0,[dB(b),gB(b)]0, ugb(b)=vdB(b), dA(a)v=gA(a)u, f(m)gB(b)=h(m)dB(b), dA(a)h(m) gA(a)f(m).Theorem3.1Let T=(OABM) be a triangular algebra which consists of A, M, B and D be a derivation of Τ, satisfies generalized Engel conditon [[???[[D (Xm), Xn1], Xn2],·],Xnk]=0, then D=0, where m, n1, n2,···nk be any positive integers.Theorem3.2Let T=(OAMB) be a triangular algebra which consists of A, M, B and D, G be derivations of T, satisfies generalized Engel conditon [[···[[D(Xm)Xn XpG(Xq),Xn1],Xn2],···],Xnk]=0, thenD=G=0, where m, n, p, q, n1n2,??? nk be any positive integers.Theorem3.3Let T=(OABM) be a triangular algebra which consists of A, M, B and D, G be derivations of T, satisfes Engel conditon [D(X), X]kX—X[G(X), X]k=0, then D=G=0. Theorem4.1Let Τ=(OABM)be a triangular algebra which consists of A, M, B and D be derivation of T, then the derivation D will not be strong commutativity preserving.Theorem4.2Let Τ=(OABM) be a triangular algebra which consists of A, M, B and D be a derivation of T, satisfies the generalized identity [D(X), D(Y)]=0, if and only if dA{A)2=0, dB{B)2=0, dA{A)u=udB{B)=0, dA{A)f{M)=f(M)dB(B)=0...