In mathematics, a **clean ring** is a ring in which every element can be written as the sum of a unit and an idempotent. A ring is a local ring if and only if it is clean and has no idempotents other than 0 and 1. The endomorphism ring of a continuous module is a clean ring.^{[1]} Every clean ring is an exchange ring.^{[2]} A matrix ring over a clean ring is itself clean.^{[3]}