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, 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