Constants of Algebraic Derivations in Prime Rings

Abstract

Let R be a prime ring with extended centroid F and δ an F-algebraic derivation of R with the associated inner derivation ad(b). Set , the subring of constants of δ in R. We characterize the primeness and semiprimeness of R ^(δ) by the minimal polynomial μ(λ) of b over F. We also show that, if R ^(δ) is a prime PI-ring, then PI-deg(R) = PI-deg(R ^(δ)) × deg _F b, where PI-deg(R) denotes the PI-degree of R. This is also extended to the case when R ^(δ) is a semiprime ring.

Key Words:

• Algebraic derivation
• Extended centroid
• Inner and outer degrees
• PI-degree
• Prime ring

2000 Mathematics Subject Classification:

• 16N60
• 16K20
• 16R50

ACKNOWLEDGMENTS

The second author is grateful to the kind hospitality of Memorial University of Newfoundland when part of the work was done during his visit there. The research is supported by NSC and NCTS-TPE of Taiwan and NSERC of Canada.

Notes

†Members of Mathematics Division (Taipei Office), National Center for Theoretical Sciences.

Communicated by E. I. Zelmanov.