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On Embedding Rings in Clean Rings

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.