ON INTEGRALITY AND GOING-DOWN INSIDE THE FIXED RING OF A MONOID RING

Abstract

An example is given of a finitely generated abelian torsion-free monoid S on which the group G with two elements acts via semigroup automorphisms such that for any field K, when the given action is extended so that G acts on the monoid ring K[X; S] via ring automorphisms that fix K elementwise, the ring extension K[X; SG] ⊆ (K[X; S])G is not integral and does not satisfy the going-down property.

Keywords

• Commutative ring, ring extension, group action, fixed ring, integrality, going-down, semigroup, fixed semigroup, semigroup ring, INC, LO, factor semigroup, maximal ideal, Krull dimension, orbit, locally finite