Rigid Rings and Makar-Limanov Techniques

Abstract

A ring is rigid if it admits no nonzero locally nilpotent derivation. Although a “generic” ring should be rigid, it is not trivial to show that a ring is rigid. We provide several examples of rigid rings and we outline two general strategies to help determine if a ring is rigid, which we call “parametrization techniques.” and “filtration techniques.” We provide many tools and lemmas which may be useful in other situations. Also, we point out some pitfalls to beware when using these techniques. Finally, we give some reasonably simple rings for which the question of rigidity remains unsettled.

Key Words:

• Additive group action
• Locally nilpotent derivation
• Rigid ring

2010 Mathematics Subject Classification:

• 14R20
• 13A50

ACKNOWLEDGMENT

The authors are grateful to Gene Freudenburg for providing several useful comments.

Stefan Maubach funded Veni-grant of council for the physical sciences, Netherlands Organisation for scientific research (NWO).

*Funded by Veni-grant of council for the physical sciences, Netherlands Organisation for Scientific Research (NWO).

Notes

Communicated by J. T. Yu.