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

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