Idempotent chains and bounded generation of SL2

On the structure of semigroups whose regular elements are completely regular

“…Commutative semigroups and semigroups with central idempotents are strongly 2-chained semigroups. Strongly 2-chained semigroups also appear naturally in the study of variants semigroups (Theorem 1.3), free idempotent-generated semigroups (Example 2.10), or in ring theory [37,38,39,46,48,49]. Our main result is a semilattice decomposition of strongly 2-chained semigroups into poor ideal extensions of completely simple semigroups and idempotent-free semigroups.…”

“…It follows that chains of idempotents of a ring are easier to handle than in the semigroup case. For instance, they can be characterized using some "generalized Euclidean algorithm" [37]. Secondly, they relate to properties of direct summands of modules.…”

“…Results on these semigroups can be found in [48], which also discusses ring theoretical aspects. In fact, many recent results have been proved regarding 2, 3 and 4-chained rings [30,37,38,43,46,48,49].…”