Rings in which elements are uniquely the sum of an idempotent and a unit. (English) Zbl 1057.16007

An associative ring with a unit is called ‘clean’ if every element is the sum of an idempotent and a unit; if this representation is unique for every element, this ring will be called ‘uniquely clean’. The authors prove that a ring is uniquely clean if and only if it is Boolean modulo the Jacobson radical and idempotents lift uniquely modulo the radical. They also prove that every image of a uniquely clean ring is again uniquely clean.


16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16U60 Units, groups of units (associative rings and algebras)
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