Depth of Idempotent‐Generated Subsemigroups of a Regular Ring

If S is an idempotent‐generated semigroup, its depth is the minimum number of idempotents needed to express a general element as a product of idempotents. Here we study the depth of S where S is the semigroup generated by all the idempotents of a von Neumann regular ring R , and the depth of various subsemigroups of S . For example, if R is directly finite, the depth of S equals the index of nilpotence of R , which considerably extends a result of Ballantine (1978) for matrices over a field. We also answer a query of Professor Howie by supplying a ring‐theoretic explanation of Reynolds' and Sullivan's (1985) result that the depth is 3 for certain subsemigroups in the infinite‐dimensional full linear case.