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On Annelidan, Distributive, and Bézout Rings

Part of: Local rings and generalizations Rings and algebras arising under various constructions Boolean algebras (Boolean rings) Radicals and radical properties of rings Distributive lattices Conditions on elements Modules, bimodules and ideals Chain conditions, growth conditions, and other forms of finiteness

Published online by Cambridge University Press:  03 May 2019

Greg Marks  and
Ryszard Mazurek
Greg Marks
Affiliation:
Department of Mathematics and Statistics, St. Louis University, St. Louis, MO63103, USA Email: [email protected]
Ryszard Mazurek
Affiliation:
Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A, 15–351Białystok, Poland Email: [email protected]
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Abstract

A ring is called right annelidan if the right annihilator of any subset of the ring is comparable with every other right ideal. In this paper we develop the connections between this class of rings and the classes of right Bézout rings and rings whose right ideals form a distributive lattice. We obtain results on localization of right annelidan rings at prime ideals, chain conditions that entail left-right symmetry of the annelidan condition, and construction of completely prime ideals.

Keywords

annelidan ringdistributive ringBézout ring

MSC classification

Primary: 06D99: None of the above, but in this section
Secondary: 16D25: Ideals 16P50: Localization and Noetherian rings 16P70: Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence 16S85: Rings of fractions and localizations 16U20: Ore rings, multiplicative sets, Ore localization 16L30: Noncommutative local and semilocal rings, perfect rings 16N20: Jacobson radical, quasimultiplication 16N60: Prime and semiprime rings 16P40: Noetherian rings and modules 16P60: Chain conditions on annihilators and summands
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 4 , August 2020 , pp. 1082 - 1110
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

The second author was supported by Polish KBN Grant 1 P03A 032 27. Parts of this paper were written while the first author was visiting the Bialystok University of Technology; other parts were written while the second author was visiting St. Louis University. Each is deeply grateful for the warm hospitality extended by both institutions.

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