Rings with internal cancellation.
The paper is an exhaustive study of the class of rings that satisfy internal cancellation with respect to their one-sided ideals. The authors propose the name of IC rings for such rings and give various new characterizations of them in terms of suitable stable range conditions as well as unique generator properties of idempotent generated right ideals. A ring \(R\) is said to be of regular stable range 1 if, for any regular elements \(a,b\in R\), \(aR+bR=R\) implies that \(a+bx\) is a unit of \(R\). An element \(a\in R\) is said to have the right unique generator property if, for any \(b\in R\), \(aR=bR\) implies that \(b=au\) for some unit \(u\) of \(R\). Then it is shown that a ring is IC if and only if it has regular stable range 1 if and only if all idempotents (or equivalently, all regular (resp. unit-regular) elements) have the right unique generator property. The authors study the functorial behavior of IC rings and characterize the rings whose matrix rings are IC as the rings whose finitely generated projective modules satisfy cancellation. Two elements \(a,b\in R\) are called similar if \(a=u^{-1}bu\) for some unit \(u\) of \(R\), and pseudo-similar if there exist \(x,y,z,w\in R\) such that \(a=zbx\), \(b=xaw\) and \(x=xzx=xwx\). It is shown that a regular element \(x=xyx\in R\) is unit-regular if and only if there exists a unit \(u\) of \(R\) such that \(xy=xu\) and \(yx=ux\). The authors use this result to study pseudo-similarity and prove that IC rings are precisely those rings for which similarity and pseudo-similarity are the same. The paper concludes with a treatment of ideals for which idempotents lift modulo all one-sided subideals and with an appendix by R. G. Swan on the failure of cancellation for finitely generated projective modules over complex group algebras.
scientific article; zbMATH DE number 2136971
scientific article