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On Annelidan, Distributive, and Bézout Rings

Part of:Distributive latticesLocal rings and generalizationsRadicals and radical properties of ringsRings and algebras arising under various constructionsModules, bimodules and idealsBoolean algebras (Boolean rings)Chain conditions, growth conditions, and other forms of finitenessConditions on elements

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: marks@slu.edu
Ryszard Mazurek
Affiliation:
Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A, 15–351Białystok, Poland Email: r.mazurek@pb.edu.pl
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Abstract

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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: Ideals16P50: Localization and Noetherian rings16P70: Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence16S85: Rings of fractions and localizations16U20: Ore rings, multiplicative sets, Ore localization16L30: Noncommutative local and semilocal rings, perfect rings16N20: Jacobson radical, quasimultiplication16N60: Prime and semiprime rings16P40: Noetherian rings and modules16P60: Chain conditions on annihilators and summands

Information

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

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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