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Spring 2020 Kronecker function rings and power series rings
Gyu Whan Chang
J. Commut. Algebra 12(1): 27-51 (Spring 2020). DOI: 10.1216/jca.2020.12.27

Abstract

Let D be an integral domain with quotient field K , X be an indeterminate over D , D [ [ X ] ] be the power series ring over D , and c ( f ) be the ideal of D generated by the coefficients of f D [ [ X ] ] . We will say that a star operation on D is a c-star operation if (i) c ( f g ) = ( c ( f ) c ( g ) ) for all 0 f , g D [ [ X ] ] and (ii) ( A B ) ( A C ) implies B C for all nonzero fractional ideals A , B , C of D . Assume that D admits a c-star operation , and let Kr ( ( D , ) ) = { f g f , g D [ [ X ] ] , g 0 , and  c ( f ) c ( g ) } . Among other things, we show that Kr ( ( D , ) ) is a Bézout domain, D is completely integrally closed, the v -operation on D is a c-star operation, and Kr ( ( D , v ) ) is a completely integrally closed Bézout domain. We also show that if V is a rank-one valuation domain, then the v -operation on V is a c-star operation, Kr ( ( V , v ) ) is a rank-one valuation domain, and Kr ( ( V , v ) ) is a DVR if and only if V is a DVR. Using this result, we show that if D is a generalized Krull domain, then Kr ( ( D , v ) ) is a one-dimensional generalized Krull domain.

Citation

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Gyu Whan Chang. "Kronecker function rings and power series rings." J. Commut. Algebra 12 (1) 27 - 51, Spring 2020. https://doi.org/10.1216/jca.2020.12.27

Information

Received: 1 January 2017; Revised: 2 June 2017; Accepted: 18 June 2017; Published: Spring 2020
First available in Project Euclid: 13 May 2020

zbMATH: 07211323
MathSciNet: MR4097054
Digital Object Identifier: 10.1216/jca.2020.12.27

Subjects:
Primary: 13A15, 13F05, 13F25, 13F30

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium

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Vol.12 • No. 1 • Spring 2020
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