COMMUTATIVE RINGS WHOSE ELEMENTS ARE A SUM OF A UNIT AND IDEMPOTENT

ABSTRACT

As defined by Nicholson a (noncommutative) ring is a clean ring if every element of is a sum of a unit and an idempotent. Let be a commutative ring with identity. We define to be a uniquely clean ring if every element of can be written uniquely as the sum of a unit and an idempotent. Examples of clean rings (uniquely clean rings) include von Neumann regular rings (Boolean rings) and quasilocal rings (with residue field ). A ring is a clean ring or uniquely clean ring if and only if is. So every zero-dimensional ring is a clean ring, but a zero-dimensional ring is a uniquely clean ring if and only if is a Boolean ring.

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