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]


  1. ^ Camillo, V.P.; Khurana, D.; Lam, T.Y.; Nicholson, W.K.; Zhou, Y. (October 2006). "Continuous modules are clean". Journal of Algebra. 304 (1): 94–111. doi:10.1016/j.jalgebra.2006.06.032.
  2. ^ Nicholson, W. K. (1977). "Lifting idempotents and exchange rings" (PDF). Transactions of the American Mathematical Society. 229: 269–278. doi:10.1090/S0002-9947-1977-0439876-2. MR 0439876. Retrieved 9 June 2016.
  3. ^ Hana, Juncheol; Nicholson, W. K. (2001). "Extensions of Clean Rings". Communications in Algebra. 29 (6): 2589–2595. doi:10.1081/AGB-100002409. S2CID 122957451.