Valuations and Prufer Rings

In this paper we consider five possible extensions of the Prufer domain notion to the case of commutative rings with zero divisors and relate the corresponding properties on a ring with the property of its total ring of quotients. We show that a Prufer ring R satisfies one of the five conditions if and only if the total ring of quotients Q(R) of R satisfies that same condition. We focus in particular on the Gaussian property of a ring

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“…We proceed by developing several notions that will be used in our proofs, and recalling some definitions found in [10], [11], [12] or [15].…”

Section: The Total Ring Of Quotientsmentioning

confidence: 99%

“…(See [10,11].) Let P be a prime ideal of R. The large quotient ring of R with respect to P , denoted by R [P ] , consists of the elements x ∈ Q(R) such that xs ∈ R for some element s ∈ R \ P .…”

Section: Definitionmentioning

confidence: 99%

Gaussian properties of total rings of quotients

2007

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“…Introduction, preliminaries. We shall extend results of Samuel [19] and Griffin [8,9] about conditions which generalise the notion of valuation domain in a field. Let U be a commutative ring with identity, R a subring of U and L an /?-submodule of U.…”

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confidence: 67%

“…Next, when R is (CM) we aim to associate with it two (BV) overrings (see The equivalence of (2) and (3) in Theorem 4.9 is closely related, when R is (MV), to [16,Proposition 3] and [9,Proposition 4]. THEOREM We end with a discussion on cancellative evaluations.…”

Section: Suppose That R Is (Cm) and Denote By P Its (Cm) Ideal P+(rmentioning

confidence: 99%

On valuation subalgebras and their centres

Rhodes

1989

Glasgow Math. J.

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1. Introduction, preliminaries. We shall extend results of Samuel [19] and Griffin [8,9] about conditions which generalise the notion of valuation domain in a field. Let U be a commutative ring with identity, R a subring of U and L an /?-submodule of U. The conditions we study have in common the property (EV), that the submodules L:x (x e U) form a chain. We pay particular attention to the strongest of the conditions, viz, that L be a Manis valuation (MV) subring, i.e. having a prime ideal P such that (L, P) is a maximal pair in U (see [19], [16] and e.g. [4]). Such P is unique, being the union of all L:x such that x $ L, which we call P+{L) the centre of L. This set P+ plays a key role in the study of all our valuation conditions.The main definitions are in Section 1. In Section 2 basic results on localisation and factoring by a (/-ideal are followed by applications. Samuel's result [19, Theoreme 5] that if (R, P) is (MV) then R satisfies the Bourbaki condition (BV), defined in Section 1.2, is extended to the case where P is replaced by a finite chain of prime ideals. In Section 3 we show that if R is (BV) then prime R-ideals containing P+{R) are also (BV). The method gives new criteria for a submodule to be (BV). In Section 4, when L is (EV), we discuss the evaluation map v L given by v L (x) = L:x. Our main interest is in the cases L = P+(R), used in [16] to characterise (MV) rings, and L = R mentioned in [8]. At the end we discuss the relation, given in [8], between (BV) rings and evaluations with cancellative image.

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“…The purpose of this paper is to prove an approximation theorem for V-topologies on (not necessarily commutative) rings along the lines of the approximation theorem for valuations in [8], [9], [10], [11]. The result is valid for a certain class of rings called rings with enough units, and a certain class of V-topologies called coarse V-topologies.…”

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confidence: 99%