Krull Properties of Semigroup Rings

Abstract

ΣaαXα for aα∈D and α∈G almost all aα are zero. Various algebraic properties have been studied by various authors ([1],[3],[4],[5],[8],[9],[10],[11],[12], [13],[14],[15],[16] etc.). We concern Krull properties of the group ring D[X;G] in this paper. Let K be the quotient field of D and let F={Vλ;λ ∈ Λ} be a set of valuation rings of k. We concern following properties on F: (E1) D=∩{Vλ;λ ∈Λ}; (E2) Each Vλ is rank 1 discrete; (E2)' Each Vλ has rank 1; (E2)" Each Vλ is a rational number valued valuation ring; (E3) F has finite character-that is, if 0≠x∈K, then x is a nonunit in only finitely many of the valuation rings in F;