# Clean ring

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]

## References

- 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. - "Lifting idempotents and exchange rings" (PDF).
*American Mathematical Society*. Retrieved 9 June 2016. - Hana, Juncheol; Nicholson, W. K. (2001). "EXTENSIONS OF CLEAN RINGS".
*Communications in Algebra*.**29**(6): 2589–2595. doi:10.1081/AGB-100002409.

This article is issued from
Wikipedia.
The text is licensed under Creative
Commons - Attribution - Sharealike.
Additional terms may apply for the media files.