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1989•Canadian Mathematical BulletinOpen access

On Condensed Noetherian Domains Whose Integral Closures are Discrete Valuation Rings

Christian Gottlieb

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Abstract

Abstract A condensed domain is an integral domain such that IJ = {xy : x ∊ I, y ∊ J } holds for each pair I, J of ideals. We prove that, under suitable conditions, a subring of a discrete valuation ring is condensed if and only if it contains an element of value 2. We also define the concept strongly condensed.

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Abstract A condensed domain is an integral domain such that IJ = {xy : x ∊ I, y ∊ J } holds for each pair I, J of ideals. We prove that, under suitable conditions, a subring of a discrete valuation ring is condensed if and only if it contains an element of value 2. We also define the concept strongly condensed.

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Abstract A condensed domain is an integral domain such that IJ = {xy : x ∊ I, y ∊ J } holds for each pair I, J of ideals. We prove that, under suitable conditions, a subring of a discrete valuation ring is condensed if and only if it contains an element of value 2. We also define the concept strongly condensed.

Key concepts: Subring, Noetherian, Mathematics, Integral domain, Discrete valuation, Discrete valuation ring, Valuation (finance), Pure mathematics

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