Genelleştirilmiş Türevli Yarıasal Halkaların Lie İdealleri

Öz

R, 2-torsion free bir yarıasal halka ve U, R halkasının bir merkez tarafından kapsanılmayan kare-kapalı Lie ideali olsun. Eğer her x,y∈R için F(xy) = F(x)y + xd(y), koşulunu sağlayan bir d:R→R türevi varsa F dönüşümüne R halkasının d ile belirlenmiş bir genelleştirilmiş türevi denir. Bu çalışmada, aşağıdaki koşullardan biri sağlanırsa d dönüşümünün U üzerinde komüting dönüşüm olduğu gösterilecektir: i) F(u)u = ±uG(u), ii) [F(u),v] = ±[u,G(v)], iii) F(u)∘v = ±u∘G(v), iv) [F(u),v] = ±u∘G(v), v) F([u,v]) = [F(u),v] + [d(v),u]. Burada G:R→R dönüşümü h:R→R türevi ile belirlenmiş bir genelleştirilmiş türevdir.

Anahtar Kelimeler

• Yarıasal halka,Lie ideal,Türev,Genelleştirilmiş türev

Lie Ideals of Semiprime Rings with Generalized Derivations

Öz

Let R be a 2- torsion free semiprime ring, U a noncentral square-closed Lie ideal of R. A map F:R→R  is called a generalized derivations if there exists a derivation d:R→R such that F(xy)=F(x)y+xd(y), for all x,y∈R. In the present paper, we shall prove that h is commuting map on U if any one of the following holds: i) F(u)u=±uG(u), ii) [F(u),v]=±[u,G(v)], iii) F(u)∘v=± u∘G(v), iv) [F(u),v]=±u∘G(v), v)F([u,v])=[F(u),v]+[d(v),u] for all u,v∈U, where G:R→R  is a generalized derivation associated with the derivation h:R→R.

Anahtar Kelimeler

• semiprime ring,Lie ideal,derivation,generalized derivation

Kaynakça

1. R. Awtar, Lie structure in prime rings with derivations, Publ. Math. Debrecen 31, 1984, 209-215.
2. J. Bergen, I. N. Herstein, W. Kerr, Lie ideals and derivation of prime rings, J. of Algebra 71, 1981, 259-267.
3. M. Bresar, On the distance of the composition of two derivations to the generalized derivations, Glasgow Math. J. 33, 1991, 89-93.
4. M. Bresar, On skew-commuting mappings of rings, Bull. Austral. Math. Soc. 47, 1993, 291--296.
5. N. Divinsky, On commuting automorphisms of rings, Trans. Roy. Soc. Canada Sect. III. 49, 1955, 19-52.
6. M. Hongan, N. Rehman, R. M. Al-Omary, Lie ideals and Jordan triple derivations in rings: Rend. Semin. Mat. Univ. Padova, 125, 2011, 147--156.
7. Ö. Gölbaşı, E. Koç, Generalized derivations on Lie ideals in prime rings: Turk. J. Math., 35, 2011, 23-28.
8. P. H. Lee, T. K. Lee, Lie ideals of prime rings with derivations, Bull. Institute of Math. Acedemia Sinica, 11, 1983, 75-79.

9. E. C. Posner, Derivations in prime rings, Proc Amer. Math. Soc. 8, 1957, 1093-1100.
10. N. Rehman, M. Hongan, Generalized Jordan derivations on Lie ideals associate with Hochschild 2-cocycles of rings, Rend. Circ. Mat. Palermo 60 (3), 2011, 437-444.
11. J. Vukman, Identities with derivations and autommorphisms on semiprime rings, Internat J. Math. and Math. Sci. 2005 (7), 2005, 1031-1038.