TAILIEUCHUNG - Note on generalized jordan derivations associate with Hochschild 2-cocycles of rings

We introduce a new type of generalized derivations associate with Hochschild 2-cocycles and prove that every generalized Jordan derivation of this type is a generalized derivation under certain conditions. This result contains the results of I. N. Herstein and M. Ashraf and N-U. Rehman. | Turk J Math 30 (2006) , 403 – 411. ¨ ITAK ˙ c TUB Note on Generalized Jordan Derivations Associate with Hochschild 2-cocycles of Rings∗ Atsushi Nakajima Abstract We introduce a new type of generalized derivations associate with Hochschild 2-cocycles and prove that every generalized Jordan derivation of this type is a generalized derivation under certain conditions. This result contains the results of I. N. Herstein [6, Theorem ] and M. Ashraf and N-U. Rehman [1, Theorem]. Key Words: Derivation, Jordan derivation, generalized derivation, generalized Jordan derivation, Hochschild 2-cocycle . 1. Introduction Let R be a ring and let x, y be arbitrary elements of R. M is called an R-bimodule if M is a left and a right R-module such that x(my) = (xm)y for all m ∈ M . Let f : R → M be an additive map. f is called a generalized derivation if there exists a derivation d : R → M such that f(xy) = f(x)y + xd(y). (1) and f is called a generalized Jordan derivation if there exists a Jordan derivation J : R → M such that (2) f(x2 ) = f(x)x + xJ(x), We denote (1) and (2) by (f, d) and (f, J), respectively. These types of generalized derivations were introduced by M. Breˇsar [2] and their properties have been discussed in ∗ Dedicated to Professor S ¸ erif Yenig¨ ul on his 60th birthday 403 NAKAJIMA many papers. In [7], another type of generalized derivations was defined by the author as follows. f is called a generalized derivation if there exists an element ω ∈ M such that f(xy) = f(x)y + xf(y) + xωy, (3) and f is called a generalized Jordan derivation if f(x2 ) = f(x)x + xf(x) + xωx, (4) which denote by (f, ω). Some categorical properties of these generalized derivations were given in [7]. In the case R has an identity element 1, if (f, d) is a generalized derivation of type (1), then (f, −f(1)) is a generalized derivation of type (3); and conversely, if (f, ω) is a generalized derivation of type (3), then f + ω` : R → M is a derivation and (f, f + ω` ) is a

• Toán học
• Derivation associate
• Jordan derivation
• Generalized derivation
• Generalized Jordan derivation
• Hochschild 2 cocycle