Idempotent chains and bounded generation of \(\mathrm{SL}_2\) (Q6103348)

The article under review is concerned with the examination of connections between association chains of idempotents and bounded generation of the special linear group \(\mathrm{SL}_2\) by elementary matrices. Thus, it is allowed to identify certain classes of matrix rings that have universal bounds on the length of association chains. In particular, many matrix rings have the exotic property that every regular element is special clean. The authors also improved the usual criterion for checking perspectivity via association chains. The main results, which are claimed to be presently unknown (in despite of Theorem 3.1 which has given a new proof), are stated and proved in Theorems 3.2, 3.8, 3.12 and 4.3, respectively. The proofs are, in all, not so difficult and use standard techniques.