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On Embedding Rings in Clean Rings

Title:

On Embedding Rings in Clean Rings

Burgess, W. D. and Raphael, Robert (2013) On Embedding Rings in Clean Rings. Communications in Algebra, 41 (2). pp. 552-564. ISSN 0092-7872

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Official URL: http://dx.doi.org/10.1080/00927872.2011.607873

Abstract

A clean ring is one in which every element is a sum of an idempotent and a unit. It is shown that every ring can be embedded in a clean ring as an essential ring extension. It is seen that the centre of a clean ring need not be a clean ring. There is no “clean hull” of a ring. A family of examples is given where there is a ring R, not a clean ring, embedded in a commutative clean ring S so that there is no clean ring T, R T S, minimal with that property. It is also shown that a commutative pm ring (each prime ideal is contained in a unique maximal ideal) cannot be extended to a clean ring by the adjunction of finitely many central idempotents.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Article
Refereed:Yes
Authors:Burgess, W. D. and Raphael, Robert
Journal or Publication:Communications in Algebra
Date:2013
Digital Object Identifier (DOI):10.1080/00927872.2011.607873
Keywords:Clean ring, Local ring, Rings of continuous functions
ID Code:976886
Deposited By: DANIELLE DENNIE
Deposited On:14 Feb 2013 19:00
Last Modified:18 Jan 2018 17:43

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