Journal article
On S -GCD domains
Journal of Algebra and its Applications, Vol.18(4), 1950067
2019
DOI: 10.1142/S0219498819500671
Abstract
Let S be a multiplicative set in an integral domain D. A nonzero ideal I of D is said to be S-v-principal if there exist an s S and a D such that sI E aD E Iv. Call D an S-GCD domain if each finitely generated nonzero ideal of D is S-v-principal. This notion was introduced in [A. Hamed and S. Hizem, On the class group and S-class group of formal power series rings, J. Pure Appl. Algebra 221 (2017) 2869-2879]. One aim of this paper is to characterize S-GCD domains, giving several equivalent conditions and showing that if D is an S-GCD domain then DS is a GCD domain but not conversely. Also we prove that if D is an S-GCD S-Noetherian domain such that every prime w-ideal disjoint from S is a t-ideal, then D is S-factorial and we give an example of an S-GCD S-Noetherian domain which is not S-factorial. We also consider polynomial and power series extensions of S-GCD domains. We call D a sublocally s-GCD domain if D is a {s n |n N}-GCD domain for every non-unit s D\{0} and show, among other things, that a non-quasilocal sublocally s-GCD domain is a generalized GCD domain (i.e. for all a,b D\{0},aD U bD is invertible). © 2019 World Scientific Publishing Company.
Details
- Title: Subtitle
- On S -GCD domains
- Creators
- D.D. Anderson - University of IowaA. Hamed - Department of Mathematics, Faculty of Sciences, Monastir, TunisiaM. Zafrullah - Idaho State University
- Resource Type
- Journal article
- Publication Details
- Journal of Algebra and its Applications, Vol.18(4), 1950067
- Publisher
- World Scientific Publishing Co. Pte Ltd
- DOI
- 10.1142/S0219498819500671
- ISSN
- 0219-4988
- Language
- English
- Date published
- 2019
- Academic Unit
- Mathematics
- Record Identifier
- 9984230626202771
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