## Differential identities in prime rings with involution

HTML articles powered by AMS MathViewer

- by Charles Lanski PDF
- Trans. Amer. Math. Soc.
**291**(1985), 765-787 Request permission

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.## References

- 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

## Additional Information

- © 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