RINGS IN WHICH ELEMENTS ARE UNIQUELY THE SUM OF AN IDEMPOTENT AND A UNIT
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Abstract algebra result on uniquely clean rings; pure mathematics.
This is a mathematical study of ring structures, not research practice or the research system.
Pure ring theory in algebra; mathematics of clean rings, not a study of research practice.
Abstract
An associative ring with unity is called clean if every element is the sum of an idempotent and a unit; if this representation is unique for every element, we call the ring uniquely clean . These rings represent a natural generalization of the Boolean rings in that a ring is uniquely clean if and only if it is Boolean modulo the Jacobson radical and idempotents lift uniquely modulo the radical. We also show that every image of a uniquely clean ring is uniquely clean, and construct several noncommutative examples.
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The record
- Venue
- Glasgow Mathematical Journal
- Topic
- Rings, Modules, and Algebras
- Field
- Mathematics
- Canadian institutions
- Memorial University of NewfoundlandUniversity of Calgary
- Funders
- Natural Sciences and Engineering Research Council of CanadaUniversity of Calgary
- Keywords
- MathematicsIdempotenceModuloUnit (ring theory)Ring (chemistry)Element (criminal law)Lift (data mining)GeneralizationBoolean ringAssociative propertyPure mathematicsCombinatoricsPrincipal ideal ringDiscrete mathematicsCommutative ringCommutative propertyComputer scienceChemistry
- Has abstract in OpenAlex
- yes