Ring $R_{ 70 }$

Bergman's exchange ring that isn't clean


Construction described on the expanded details page for this ring.

Notes: Description is Šter's construction of the opposite ring of Bergman's original. It has been said that all examples of nonclean exchange rings so far are based on this one.

Expanded details page.

Keywords infinite matrix ring quotient ring subring


  • D. Handelman. Perspectivity and cancellation in regular rings. (1977) @ Example 1
  • J. Ster. Corner rings of a clean ring need not be clean. (2012) @ Example 3.1
  • J. Ster. Lifting units in clean rings. (2013) @ Bergman's example
  • K. R. Goodearl. Von Neumann regular rings. (1991) @ Example 4.26, pp 46-48
  • T.-K. Lee, Z. Yi, and Y. Zhou. An example of Bergman's and the extension problem for clean rings. (2008) @ Bergman's example
  • V. P. Camillo and H.-P. Yu. Exchange rings, units and idempotents. (1994) @ p 4746

  • = has the property
  • = does not have the property
  • = information not in database
Name Measure
composition length left: $\infty$right: $\infty$
weak global dimension 0
Name Description
Jacobson radical $\{0\}$
Left singular ideal $\{0\}$
Right singular ideal $\{0\}$