Generalized Jordan triple derivations associated with Hochschild 2–cocycles of rings

Let R denote an associative ring with center Z(R). A ring R is said to have characteristic n if n is the least positive integer such that nx=0 for all x∈R, and of characteristic not n if nx=0, x∈R, then x=0. An additive subgroup L of R is called a Lie ideal of R if [u,r]∈L for all u∈L, r∈R. A Lie ideal L is said to be a square-closed Lie ideal of R if u²∈L for all u∈L. An R-bimodule M is a left and right R-module such that x(my)=(xm)y for all m∈M and x,y∈R. Recall that a ring R is called prime if xRy=(0) implies that either x=0 or y=0, and R is called semiprime if xRx=(0) implies x=0. An additive mapping d:R→R is called a derivation if d(xy)=d(x)y+xd(y) for all x,y∈R. d is called a Jordan derivation in case d(x²)=d(x)x+xd(x) for all x∈R. Moreover, d is called a Jordan triple derivation if d(xyx)=d(x)yx+xd(y)x+xyd(x) for all x,y∈R. It is obvious to see that every derivation is a Jordan derivation and is a Jordan triple derivation but the converse is in general not true. A classical result of Herstein [1] asserts that any Jordan derivation of a prime ring with characteristic different from 2 is a derivation. In [2], Bre\(\breve {s}\)ar has proved Herstein’s result in the case of a semiprime ring. Also, he has shown in [3] that any Jordan triple derivation of a 2-torsion free semiprime ring is a derivation. An additive map f of a ring R is called a generalized derivation if there is a derivation d of R such that for all x, y in R, f(xy)=f(x)y+xd(y) and is called a generalized Jordan derivation if there is a Jordan derivation d such that f(x²)=f(x)x+xd(x) for all x∈R. Furthermore, f is said to be a generalized Jordan triple derivation if there is a Jordan triple derivation d of R such that for all x, y in R, f(xyx)=f(x)yx+xd(y)x+xyd(x). In [4], Jing and Lu have proved in a prime ring R of characteristic not two that every generalized Jordan derivation of R is a generalized derivation, and also every generalized Jordan triple derivation on R is a generalized derivation.

Let θ and ϕ be endomorphisms of a ring R. f is called a (θ,ϕ)−derivation if f(xy)=f(x)θ(y)+ϕ(x)f(y) for all x, y∈R. f is called a Jordan (θ,ϕ)–derivation if f(x²)=f(x)θ(x)+ϕ(x)f(x) for all x∈R. f is called a Jordan triple (θ,ϕ)−derivation if f(xyx)=f(x)θ(y)θ(x)+ϕ(x)f(y)θ(x)+ϕ(x)ϕ(y)f(x) for all x,y∈R. In [5], Liu and Shiue have proved that every Jordan triple (θ,ϕ)−derivation on a 2-torsion free semiprime ring R is a (θ,ϕ)−derivation, where θ and ϕ are automorphisms. An additive mapping f:R→R is said to be a left (right) centralizer, if f(xy)=f(x)y(f(xy)=xf(y)) for all x,y∈R. f is called a centralizer, if f is both a left and right centralizer. In [6], Vukman and Kosi-Ulbl have shown that if R is a 2-torsion free semiprime ring and f is an additive mapping of R such that 2f(xyx)=f(x)yx+xyf(x) for all x,y∈R, then f is a centralizer.

An additive mapping f:R→R is said to be a left (right) θ−centralizer associated with a function θ of R, if f(xy)=f(x)θ(y)(f(xy)=θ(x)f(y)) for all x,y∈R. f is called a θ−centralizer, if f is both a left and right θ−centralizer. Daif, El-Sayiad, and Muthana in [7] have proved that if R is a 2−torsion free semiprime ring and f is an additive mapping of R such that 2f(xyx)=f(x)θ(yx)+θ(xy)f(x) for all x,y∈R with θ(Z(R))=Z(R), where θ is a nonzero surjective endomorphism on R, then f is a θ−centralizer.

Now let R be a ring and M be an R-bimodule. A biadditive map α:R×R→M is called a Hochschild 2–cocycle, if xα(y,z)−α(xy, z)+α(x,yz)−α(x, y)z=0 for all x,y,z∈R, and α is called symmetric if α(x,y)=α(y,x) for all x,y∈R. Nakajima [8] has introduced a new type of generalized derivations and generalized Jordan derivations associated with Hochschild 2–cocycles in the following way. An additive map f:R→M is called a generalized derivation associated with a Hochschild 2–cocycle α if f(xy)=f(x)y+xf(y)+α(x,y) for all x,y∈R, and f is called a generalized Jordan derivation associated with α if f(x²)=f(x)x+xf(x)+α(x,x) for all x∈R. If α=0, then f means the usual derivation and Jordan derivation. He has given the following examples:

(1) If f is a generalized derivation associated with a derivation d, then the map α₁:R×R∋(x,y)↦x(d−f)(y)∈M is biadditive and satisfies the 2–cocycle condition. Hence, f is a generalized derivation associated with α₁.

(2) If f:R→M is a left centralizer, then by f(xy)=f(x)y+xf(y)+x(−f)(y), we have a 2–cocycle α₂:R×R→M defined by, α₂(x,y)=x(−f)(y), and hence, f is a generalized derivation associated with α₂.

(3) Let f be a (θ,ϕ)−derivation. Then, the map α₃:R×R∋(x,y)↦f(x)(θ(y)−y)+(ϕ(x)−x)f(y)∈M, is biadditive and satisfies the 2–cocycle condition. Since f(xy)=f(x)y+xf(y)+α₃(x,y), then f is a generalized derivation associated with α₃.

(4) In general, he has mentioned the following. Let f:R→M be an additive map and let α:R×R→M be a biadditive map. If f(xy)=f(x)y+xf(y)+α(x,y) holds, then by the associativity f((xy)z)=f(x(yz)), α satisfies the 2–cocycle condition. Thus f is a generalized derivation associated with α.

In the present article, we introduce the notion of generalized Jordan triple derivations associated with Hochschild 2–cocycles in the following way. Let R be a ring and let M be an R-bimodule. An additive map f:R→M is called a generalized Jordan triple derivation associated with a Hochschild 2–cocycle α if f(xyx)=f(x)yx+xf(y)x+α(x,y)x+xyf(x)+α(xy,x) for all x,y∈R.

Examples (i) If f is a Jordan triple derivation, then the zero map α₁ is biadditive and satisfies the 2–cocycle condition. Therefore f is a generalized Jordan triple derivation associated with α₁.

(ii) If f is a generalized Jordan triple derivation associated with a Jordan triple derivation d, then α₂(x,y)=x(d−f)(y) is biadditive and satisfies the 2–cocycle condition and we can see that f(xyx)=f(x)yx+xf(y)x+α₂(x,y)x+xyf(x)+α₂(xy,x). Hence f is a generalized Jordan triple derivation associated with α₂.

Our aim in this work is to show that every generalized Jordan triple derivation associated with a Hochschild 2–cocycle α from a prime ring R with characteristic different from 2 to an R-bimodule M is a generalized derivation associated with α.