Note on g-monoids

Abstract

We study almost pseudo-valuation semigroups S, especially will study semistar operations on S, and will determine the complete integral closure of S. We will study various cancellation properties of semistar operations on g-monoids. Also, we will study Kronecker function rings of any semistar operations on gmonoids. A. Badawi and E. Houston [BH] introduced an almost pseudo-valuation domain. An integral domain D with quotient field K is called an almost pseudo-valuation domain (or, an APVD) if every prime ideal P of D is strongly primary, that is, if, for elements x, y ∈ K, xy ∈ P and x 6∈ P implies y n ∈ P for some positive integer n. In this paper we will introduce an almost pseudo-valuation semigroup (or, an APVS), and will study it, especially will study semistar operations on an APVS, and will determine the complete integral closure of an APVS. Let G be a torsion-free abelian additive group. A subsemigroup S of G which contains 0 is called a grading monoid (or, a g-monoid). We may confer [M3] for g-monoids. Also, we will study various cancellation properties of semistar operations on g-monoids. Moreover, we will study Kronecker function rings of any semistar operations on g-monoids. The paper consists of seven sections. In §1, we will introduce an APVS, and will show that [BH] holds for g-monoids. In §2, we will show a semigroup version of [KMOS], and will determine the complete integral closure of the APVS. In §3, we will give conditions for an APVS to have only a finite number of semistar operations. In §4, we will study conditions for an APVD to have only a finite number of semistar operations. In §5, we will introduce various cancellation properties of semistar operations on a g-monoid, and will show various implications of the cancellation properties. In §6, we will study results for Kronecker function rings of e.a.b. semistar operations for any semistar operations on g-monoids. §7 is an appendix. Many parts in every §1 ∼ §4 are restatements of [M7]. Since it seems that [M7] has not appeared about six years, and we refered [M7] in