Lie ideals with generalized derivations and derivations of semiprime rings

• Autorzy: Sögütcü Emine Koç , Rehman Nadeem ur , Dernek Eda

Identyfikatory

DOI

10.21008/j.0044-4413.2023.0009

• Języki publikacji: EN

Abstrakty

EN

Let R be a 2-torsion free semiprime ring, L a square-closed Lie ideal of R, ø be a derivation of R and α be an automorphism of R. We will demonstrate in this study that ø(L) = (0), and so ø is a zero map on L if any one of the following holds for all r r ∈ L: (i) ø(r)r = 0 (or r ø(r) = 0) (ii) ø(r) r + r (α (r) - r) = 0, (iii) The mapping r → ø (r) + α (r) is commuting on L. Moreover, if any one of the following are satisfied for two generalized derivations (F, ø) and (H, ξ) of R, then ø is a commuting map on L: (iv) F (r) F (s) - H(rs) ∈ Z (R), (v) F (rs) = ± H (rs), (vi) F (rs) = ±H (sr), for all r, s ∈ L.

Słowa kluczowe

EN

semiprime ring; Lie ideals; derivation; generalized derivation

• Wydawca: Wydawnictwo Politechniki Poznańskiej
• Czasopismo: Fasciculi Mathematici
• Rocznik: 2023
• Tom: nr 66
• Strony: 107--120
• Opis fizyczny: Bibliogr. 17 poz.

Twórcy

autor

Sögütcü Emine Koç

• Cumhuriyet University, Faculty of Science, Department of Mathematics, Sivas, Turkey

autor

Rehman Nadeem ur

• Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

autor

Dernek Eda

• Cumhuriyet University, Faculty of Science, Department of Mathematics, Sivas, Turkey

Bibliografia

• [1] Ashraf M., Ali A., Ali S., Some commutativity theorems for rings with generalized derivations, Southeast Asian Bull. Math., 31(2007), 415-421.
• [2] Ashraf M., Rehman N., On derivations and commutativity in prime rings, East-West J. Math., 3(1) (2001), 87-91.
• [3] Awtar R., Lie structure in prime rings with derivations, Publ. Math. Debrecen, 31(1984), 209-215.
• [4] Bergen J., Herstein I.N., Kerr W., Lie ideals and derivation of prime rings, J. of Algebra 71, (1981), 259-267.
• [5] Bresar M., On the distance of the composition of two derivations to the generalized derivations, Glasgow Math. J. 33, (1991), 89-93.
• [6] Bresar M., Centralizing mappings and derivations in prime rings, J. Algebra 156, (1993), 385-394.
• [7] Bresar M., On skew-commuting mappings of rings, Bull. Austral. Math. Soc. 47, (1993), 291-296.
• [8] Dhara B., Rehman N., Raza M.A., Lie ideals and action of generalized derivations in rings, Miskolc Math Notes, 16(2) (2015), 769-779.
• [9] Divinsky N., On commuting automorphisms of rings, Trans. Roy. Soc. Canada Sect. III. 49, (1955), 19-52.
• [10] Gölbaşi Ö., Koç E., Notes on commutativity of prime rings with generalized derivations, Commun. Fac. Sci. Ank. Series A1 58, (2009), 39-46.
• [11] Mayne J.H., Centralizing automorphisms of prime rings, Canad. Math. Bull. 19, (1976), 113-115.
• [12] Luh J., A note on commuting automorphisms of rings, Amer. Math. Monthly 77, (1970), 61-62.
• [13] Posner E.C., Derivations in prime rings, Proc. Amer. Math. Soc. 8, (1957), 1093-1100.
• [14] Quadri M.A., Khan M.S., Rehman N., Generalized derivations and commutativity of prime rings, Indian J. Pure Appl. Math., 34(2003), 1393-1396.
• [15] Rehman N., Hongan M., Generalized Jordan derivations on Lie ideals associate with Hochschild 2-cocycles of rings, Rend. Circ. Mat. Palermo, 60(3) (2011), 437-444.
• [16] Rehman N., Hongan M., Al-Omary R.M., Lie ideals and Jordan triple derivations in rings, Rend. Sem. Mat. Univ. Padova, 125(2011), 147-156.
• [17] Vukman J., Identities with derivations and autommorphisms on semiprime rings, Internat J. Math. and Math. Sci., 7(2005), 1031-103.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).