2003/01/07 by Sharon M. Clarke, Clarke, Sharon M.
Computer Science · Mathematics · #Coding theory and cryptography #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras #math.AC
paper · pdf · doi:10.48550/arxiv.math/0301046
This article consists of 11 pages
arxiv created 2003/01/07 · openalex publication_date 2003/01/07 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let D be an integral domain with quotient field K. A star-operation ⋆ on D is a closure operation A \longmapsto A^⋆ on the set of nonzero fractional ideals, F(D), of D satisfying the properties: (xD)^⋆ = xD and (xA)^⋆ = xA^⋆ for all x ∈ K^∗ and A ∈ F(D). Let \M S be a multiplicatively closed set of ideals of D. For A ∈ F(D) define A\M S = \x ∈ K | xI ⊆A, for some I ∈ \M S\. Then D\M S is an overring of D and A\M S is a fractional ideal of D\M S. Let \M S be a multiplicative set of finitely generated nonzero ideals of D and A ∈ F(D), then the map A \longmapsto A\M S is a finite character star-operation if and only if for each I ∈ \M S, Iv = D. We give an example to show that this result is not true if the ideals are not assumed to be finitely generated. In general, the map A \longmapsto A\M S is a star-operation if and only if \M S, the saturation of \M S, is a localizing GV-system. We also discuss star-operations given of the form A \longmapsto ∩ ADα, where D = ∩ Dα.