*-σ-biderivations on *-rings
Subject Areas : Statistics
1 - ssistant Professor, Department of Mathematics and Computer Sciences (Analysis Research), Hakim Sabzevari University, Sabzevar, Iran
Keywords: * -σ-دواشتقاق, *-حلقه, حلقه نیمه اول, حلقه اول, *-دواشتقاق,
Abstract :
Bresar in 1993 proved that each biderivation on a noncommutative prime ring is a multiple of a commutatot. A result of it is a characterization of commuting additive mappings, because each commuting additive map give rise to a biderivation. Then in 1995, he investigated biderivations, generalized biderivations and sigma-biderivations on a prime ring and generalized the results of derivations for them. He showed every generalized biderivation G of an ideal in a prime ring R is of the form G(x,y)=xay+ybx, where a,b are the elements of the symmetric Martindale ring of quotionts of R. Ali in 2012, studied *-derivations on a *-ring and showed that *-derivations maps to the center of ring.In this paper, we introduce *-biderivations and *-sigma-biderivations on a *-ring. Then some results obtained by Bresar and Ali generalize for these mapping to a class of *-rings. That is, each *-sigma-biderivation is characterized on a prime *-ring and show that each *-biderivation maps to the center of ring.
[1]M. Brešar, On generalized biderivations and related maps, J. Algebra, 172: 765-786 (1995)
[2] S. Ali, On generalized -derivations in *-rings, palestine J. Math.,1: 32-37 (2012)
[3] E. C. Posner, Derivations in primen rinegs, Proc. Amer. Math. Soc. 8: 1093-1100 (1957)
[4] M. Brešar, Centralizing mappings and derivations in prime rings, J. Algrbra 156: 385-394 (1993)
[5] C. Lanski, Differential identities, Lie ideals, and Posner’s theorems, Pacific J. Math. 134: 275-297(1988)
[6] H. E. Bell and W. S. Martindale III, Centralizing mappings of seniprime rings. Canad. Math Bull., 30(1): 92-101 (1987)
[7] L. Oukhtite,. An extension of Posner’s second theorem to rings with involution, Int. J. Modern Math., 4: 303-308 (2009)
[8] L. Oukhtite and L. Taoufiq, Some properties of derivations on rings with involution, Int. J. Modern Math., 4: 309-315 (2009)
[9] W. S. Martindale, prime rings satis fying a generalized polynomial identity, J. Aalgebra, 12: 576-589 (1969)
[10] D. Passman, Infinite crossed products, Academic press, san Diego, (1989)
