On left derivations and related mappings
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- by M. Brešar and J. Vukman PDF
- Proc. Amer. Math. Soc. 110 (1990), 7-16 Request permission
Abstract:
Let $R$ be a ring and $X$ be a left $R$-module. The purpose of this paper is to investigate additive mappings ${D_1}:R \to X$ and ${D_2}:R \to X$ that satisfy ${D_1}(ab) = a{D_1}(b) + b{D_1}(a),a,b \in R$ (left derivation) and ${D_2}({a^2}) = 2a{D_2}(a),a \in R$ (Jordan left derivation). We show, by the rather weak assumptions, that the existence of a nonzero Jordan left derivation of $R$ into $X$ implies $R$ is commutative. This result is used to prove two noncommutative extensions of the classical Singer-Wermer theorem.References
- H. E. Bell and W. S. Martindale III, Centralizing mappings of semiprime rings, Canad. Math. Bull. 30 (1987), no. 1, 92–101. MR 879877, DOI 10.4153/CMB-1987-014-x
- Frank F. Bonsall and John Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029
- M. Brešar and J. Vukman, Jordan derivations on prime rings, Bull. Austral. Math. Soc. 37 (1988), no. 3, 321–322. MR 943433, DOI 10.1017/S0004972700026927
- M. Brešar and J. Vukman, On some additive mappings in rings with involution, Aequationes Math. 38 (1989), no. 2-3, 178–185. MR 1018911, DOI 10.1007/BF01840003
- H. G. Dales, Automatic continuity: a survey, Bull. London Math. Soc. 10 (1978), no. 2, 129–183. MR 500923, DOI 10.1112/blms/10.2.129
- I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104–1110. MR 95864, DOI 10.1090/S0002-9939-1957-0095864-2
- B. E. Johnson, Continuity of derivations on commutative algebras, Amer. J. Math. 91 (1969), 1–10. MR 246127, DOI 10.2307/2373262
- Svetozar Kurepa, The Cauchy functional equation and scalar product in vector spaces, Glasnik Mat.-Fiz. Astronom. Društvo Mat. Fiz. Hrvatske Ser. II 19 (1964), 23–36 (English, with Serbo-Croatian summary). MR 171100
- Joseph H. Mayne, Centralizing automorphisms of prime rings, Canad. Math. Bull. 19 (1976), no. 1, 113–115. MR 419499, DOI 10.4153/CMB-1976-017-1
- Edward C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093–1100. MR 95863, DOI 10.1090/S0002-9939-1957-0095863-0
- A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20 (1969), 166–170. MR 233207, DOI 10.1090/S0002-9939-1969-0233207-X
- A. M. Sinclair, Jordan homomorphisms and derivations on semisimple Banach algebras, Proc. Amer. Math. Soc. 24 (1970), 209–214. MR 250069, DOI 10.1090/S0002-9939-1970-0250069-3
- Allan M. Sinclair, Automatic continuity of linear operators, London Mathematical Society Lecture Note Series, No. 21, Cambridge University Press, Cambridge-New York-Melbourne, 1976. MR 0487371
- I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955), 260–264. MR 70061, DOI 10.1007/BF01362370
- Bertram Yood, Continuous homomorphisms and derivations on Banach algebras, Proceedings of the conference on Banach algebras and several complex variables (New Haven, Conn., 1983) Contemp. Math., vol. 32, Amer. Math. Soc., Providence, RI, 1984, pp. 279–284. MR 769517, DOI 10.1090/conm/032/769517
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 110 (1990), 7-16
- MSC: Primary 16W25; Secondary 16U80, 16W10, 16W80, 46H05
- DOI: https://doi.org/10.1090/S0002-9939-1990-1028284-3
- MathSciNet review: 1028284