Graded-valuation domains

Commutative graded-$n$-coherent and graded valuation rings

“…We begin with the following result extending Theorem 1.2 in [1] to the case where rings are with zero divisors and which characterize gr-valuation rings. Theorem 4.1.…”

“…Let A be a graded ring, where G is a commutative monoid. Following [4], a proper homogeneous ideal P of A is said to be a homogeneous 2-prime ideal if whenever ab ∈ P for some a, b ∈ h(A), then either a 2 ∈ P or b 2 ∈ P. Many characterizations of gr-valuation domains are given in [1]. Now, we give a new characterization of gr-valuation domains in terms of homogeneous 2-prime ideals.…”

Different Types of G-Prime Ideals Associated to a Graded Module and Graded Primary Decomposition in a Graded Prüfer Domain

“…The theory of rings which are graded by a finitely generated abelian group came in light mainly when homogeneous coordinate rings for toric varieties were introduced in algebraic geometry [5]. The theory of graded rings and modules can be considered as an extension of the ring and module theory which has been studied by many authors (See [1], [2], [3], [5], [10], [11], [13], [14] and [15]). A comparison between global primary decomposition of coherent sheaves over a toric variety and graded primary decomposition of graded ideals is given in [15].…”

“…Graded valuation domains and graded Prüfer domains were studied in [1] and [2], where the authors assumed that the graded domains are integral domains. So our definitions of graded valuation domain and graded Prüfer domain are generalizations of [1] and [2]. We also prove the graded local-global principle of a graded module in terms of strong G-Krull prime ideals (Theorem 3.18).…”

“…To answer this question, first we introduce the notion of graded Prüfer domain as a generalization of Prüfer domain to the graded case, then we show that most of the notions of G-prime ideals associated to a graded module over a graded Prüfer domain under certain conditions coincide (Theorem 3.9 and Proposition 3.12). Graded valuation domains and graded Prüfer domains were studied in [1] and [2], where the authors assumed that the graded domains are integral domains. So our definitions of graded valuation domain and graded Prüfer domain are generalizations of [1] and [2].…”