Graded Almost Valuation Domains
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Graded Almost Valuation Domains Chahrazade Bakkari1 · Najib Mahdou2 · Abdelkbir Riffi1 Received: 14 March 2020 / Accepted: 3 June 2020 / © Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract Let be a torsionless grading monoid, R = α∈ Rα a -graded integral domain, H the set of nonzero homogeneous elements of R, K the quotient field of R0 and G0 the group of units of . We say that R is a graded almost valuation domain (gr-AVD) if for every nonzero homogeneous element x ∈ RH , there exists an integer n = n(x) ≥ 1 with x n or x −n ∈ R. In this paper, we show that R is a gr-AVD if and only if the following conditions hold. 1. 2. 3. 4.
is an almost valuation monoid, K ⊆ (RH )0 is a root extension, If α ∈ is not a unit, then for every 0 = x ∈ Rα and 0 = r ∈ R0 , r n | x n in R for some n ≥ 1, and T = α∈G0 Rα is a gr-AVD.
Keywords Graded almost valuation domains · Graded valuation domains · Almost-valuation monoids · Valuation monoids · Almost valuation domains Mathematics Subject Classification (2010) 13A02 · 13A15
1 Introduction Throughout this paper, all rings are commutativev with unity; will denote a torsionless grading monoid with quotient group and group of units G0 = ∩ −. Our aim is to generalize the concept of almost-valuation domains (AVDs) to the context of -graded integral domains and then completely characterize this generalized property as Anderson et al. Najib Mahdou
[email protected] Chahrazade Bakkari [email protected] Abdelkbir Riffi [email protected] 1
Department of Mathematics, Faculty of Science, Box 11201 Zitoune, University Moulay Ismail, Meknes, Morocco
2
Laboratory of Modeling and Mathematical Structures, Department of Mathematics, Faculty of Science and Technology of Fez, Box 2202, University S.M. Ben Abdellah Fez, Fes, Morocco
C. Bakkari et al.
did in [2] for valuation domains. The notion of AVDs was introduced by Anderson and Zafrullah in [4] as follows. An integral domain R with quotient field K, is called an AVD if for each 0 = x ∈ K, there exists an integer n = n(x) ≥ 1 with x n or x −n ∈ R. Clearly, any valuation domain is an AVD; the converse fails [6, Examples 2.20, 3.5 and 3.7]. However, the proof of [4, Theorem 5.6] showed that any AVD is quasi-local with linearly ordered prime ideals. This paper consists of two sections excluding introduction. In Section 2, we introduce the notion of graded almost-valuation domains (gr-AVDs). Among other things, we show that a nontrivially graded integral domain R = α∈ Rα is never quasi-localand so never an AVD (Proposition 2.1). It is easy to see that a gr-valuation domain R = α∈ Rα is a gr-AVD (Proposition 2.2); Examples 3.4, 3.5, 3.12 and 3.13 show that the converse fails; but a gr-AVD R = α∈ Rα must have a unique maximal homogeneous ideal (Corollary 2.5). Concerning (torsionless grading) monoids, one sees easily that a valuation monoid is an almost valuation monoid (Proposition 2.6); the converse fails (see Example 2.7). In Section 3, we com
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