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.
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