On the Drazin index of regular elements
Let \(R\) denote a ring with unity 1. An element \(a\in R\) is regular if \(a\in aRa\), and if some power of \(a\) is regular, then \(a\) is weak-regular. An element \(a\) is Drazin invertible if the three equations \(xax= x\), \(ax= xa\), \(a^kxa= a^k\) have a common solution \(x= a^D\) for some integer \(k\geq 0\). It is well-known that if \(a^D\) exists, it is unique. The minimum \(k\) fulfilling the above is the Drazin index \(i(a)\) of \(a\). The authors relate the Drazin index of \(a\) to the Drazin index of \(a^2 a^-+ 1- aa^-\). They also give an alternative characterization when considering matrices over an algebraically closed field, and close with some questions and remarks.
scientific article; zbMATH DE number 5637474