On the Commutativity of a Prime ∗-Ring with a ∗-α-Derivation

Year 2018, Volume: 2 Issue: 3, 51 - 60, 31.07.2018

Gülay Bosnalı Neşet Aydın Selin Türkmen

Abstract

Let R be a prime ∗-ring where ∗ be an involution of R, α be an automorphism of R, T be a nonzero left α-∗-centralizer on R and d be a nonzero ∗-α-derivation on R. The aim of this paper is to prove the commutativity of a ∗-ring R with the followings conditions: i) if T is a homomorphism (or an antihomomorphism) on R,ii) if d([x,y]) = 0 for all x,y ∈ R, iii) if [d(x),y] = [α(x),y] for all x,y ∈ R, iv) if d(x)◦y = 0 for all x,y ∈ R, v) if d(x◦y) = 0 for all x,y ∈ R. 

Keywords

∗-derivation, ∗-α-derivation, left ∗-centralizer, left α-∗-centralizer

References

• References [1] HERSTEIN I.N., 1976, Rings with Involutions, Chicago Univ., Chicago Press. [2] KIM K. H. and LEE Y. H., 2017, A Note on ∗-Derivation of Prime ∗-Rings, International Mathematical Forum, 12(8), 391-398. [3] REHMAN N., ANSARI A. Z. and HAETINGER C., 2013, A Note on Homomorphisims and Anti- Homomorphisims on ∗-Ring, Thai Journal of Mathematics, 11(3), 741-750. [4] POSNER E.C.,1957, Derivations in Prime Rings, Proc. Amer. Math. Soc., 8:1093-1100. [5] BRESAR M. and VUKMAN J.,1989, On Some Additive Mappings in Rings with Involution, Aequationes Math., 38, 178-185. 10 Gu¨lay BOSNALI et al. [6] HERSTEIN I.N.,1957, Jordan Derivations of Prime Rings, Proc. Amer. Math. Soc., 8(6), 11041110. [7] ZALAR B., 1991, On Centralizers of Semiprime Rings, Comment. Math. Univ. Caroline, 32(4), 609-614. [8] SALHI A. and FOSNER A., 2010, On Jordan (α,β)∗-Derivations In Semiprime Rings, Int J. Algebra, 4(3), 99-108 [9] KOC¸ E., G¨OLBASI ¨O., 2017, Results On α-∗-Centralizers of Prime and Semiprime Rings with Involution, commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 66(1), 172-178.