Two Examples in the Theory of Fixed Rings

Abstract

An example is given of an Artinian local (commutative unital) ring R and a finite group G acting on R (via ring automorphisms) such that R ^G is not an Artinian ring; in this example, |G| necessarily fails to be a unit of R. Also, an example is constructed of a ring R on which an infinite cyclic group G acts such that the ring extension R ^G  ⊆ R does not satisfy the going-down property GD; in this second example, the G-action on R is necessarily not locally finite, and it can be arranged that R is a monoid ring with any desired infinite cardinality and that R ^G is an integral domain with any (prime or 0) characteristic.

Key Words:

• Artinian ring
• Characteristic
• Factor semigroup
• Fixed ring
• Going-down
• Krull dimension
• Locally finite group action
• Monoid ring

2000 Mathematics Subject Classification:

• Primary 13A50, 13B24
• Secondary 13A15

Notes

Communicated by I. Swanson