Correction: Trans. Amer. Math. Soc. 309 (1988), 857-859.
Abstract:Let $R$ be a prime ring with involution. Using work of V. K. Kharchenko it is shown that any generalized identity for $R$ involving derivations of $R$ and the involution of $R$ is a consequence of the generalized identities with involution which $R$ satisfies. In obtaining this result, a generalization, to rings satisfying a GPI, of the classical theorem characterizing inner derivations of finite-dimensional simple algebras is required. Consequences of the main theorem are that in characteristic zero no outer derivation of $R$ can act algebraically on the set of symmetric elements of $R$, and if the images of the set of symmetric elements under the derivations of $R$ satisfy a polynomial relation, then $R$ must satisfy a generalized polynomial identity.
- S. A. Amitsur, Generalized polynomial identities and pivotal monomials, Trans. Amer. Math. Soc. 114 (1965), 210–226. MR 172902, DOI 10.1090/S0002-9947-1965-0172902-9
- Theodore S. Erickson, Wallace S. Martindale 3rd, and J. Marshall Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975), no. 1, 49–63. MR 382379
- I. N. Herstein, Rings with involution, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1976. MR 0442017
- I. N. Herstein, A theorem on derivations of prime rings with involution, Canadian J. Math. 34 (1982), no. 2, 356–369. MR 658971, DOI 10.4153/CJM-1982-023-x
- V. K. Harčenko, Differential identities of prime rings, Algebra i Logika 17 (1978), no. 2, 220–238, 242–243 (Russian). MR 541758
- Amos Kovacs, On derivations in prime rings and a question of Herstein, Canad. Math. Bull. 22 (1979), no. 3, 339–344. MR 555163, DOI 10.4153/CMB-1979-042-0
- Charles Lanski, Invariant submodules in semi-prime rings, Comm. Algebra 6 (1978), no. 1, 75–96. MR 472896, DOI 10.1080/00927877808822234
- Wallace S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576–584. MR 238897, DOI 10.1016/0021-8693(69)90029-5
- Wallace S. Martindale III, Prime rings with involution and generalized polynomial identities, J. Algebra 22 (1972), 502–516. MR 306245, DOI 10.1016/0021-8693(72)90164-0
- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 291 (1985), 765-787
- MSC: Primary 16A38; Secondary 16A12, 16A28, 16A48, 16A72
- DOI: https://doi.org/10.1090/S0002-9947-1985-0800262-9
- MathSciNet review: 800262