On derivations in prime rings and Banach algebras
HTML articles powered by AMS MathViewer
- by J. Vukman PDF
- Proc. Amer. Math. Soc. 116 (1992), 877-884 Request permission
Abstract:
Let $R$ be a ring with center $Z(R)$. A mapping $F:R \to R$ is said to be centralizing on $R$ if $[F(x),x] \in Z(R)$ holds for all $x \in R$. The main purpose of this paper is to prove the following result, which generalizes a classical result of Posner: Let $R$ be a prime ring of characteristic not 2, 3, and 5. Suppose there exists a nonzero derivation $D:R \to R$ , such that the mapping $x \rightarrowtail [[D(x),x],x]$ is centralizing on $R$ . In this case $R$ is commutative. Combining this result with some well-known deep results of Sinclair and Johnson, we generalize Yood’s noncommutative extension of the 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
- M. Brešar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1003–1006. MR 929422, DOI 10.1090/S0002-9939-1988-0929422-1
- 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
- M. Brešar and J. Vukman, On left derivations and related mappings, Proc. Amer. Math. Soc. 110 (1990), no. 1, 7–16. MR 1028284, DOI 10.1090/S0002-9939-1990-1028284-3
- J. M. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc. 53 (1975), no. 2, 321–324. MR 399182, DOI 10.1090/S0002-9939-1975-0399182-5
- 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
- Joseph H. Mayne, Ideals and centralizing mappings in prime rings, Proc. Amer. Math. Soc. 86 (1982), no. 2, 211–212. MR 667275, DOI 10.1090/S0002-9939-1982-0667275-4
- Joseph H. Mayne, Centralizing mappings of prime rings, Canad. Math. Bull. 27 (1984), no. 1, 122–126. MR 725261, DOI 10.4153/CMB-1984-018-2
- 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
- I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955), 260–264. MR 70061, DOI 10.1007/BF01362370
- Marc P. Thomas, The image of a derivation is contained in the radical, Ann. of Math. (2) 128 (1988), no. 3, 435–460. MR 970607, DOI 10.2307/1971432
- J. Vukman, Symmetric bi-derivations on prime and semi-prime rings, Aequationes Math. 38 (1989), no. 2-3, 245–254. MR 1018917, DOI 10.1007/BF01840009
- J. Vukman, Commuting and centralizing mappings in prime rings, Proc. Amer. Math. Soc. 109 (1990), no. 1, 47–52. MR 1007517, DOI 10.1090/S0002-9939-1990-1007517-3
- 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 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 116 (1992), 877-884
- MSC: Primary 46H40; Secondary 16N60, 16W25, 46H99
- DOI: https://doi.org/10.1090/S0002-9939-1992-1072093-8
- MathSciNet review: 1072093