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

Nicholson, W. K.; Zhou, Y.

Glasg. Math. J. 46, No. 2, 227-236 (2004).

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.

Reviewer: Xue Weimin (Fujian)

MSC:

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

Keywords:

uniquely clean rings; Boolean rings; idempotents; units