Adjoining a Universal Inner Inverse to a Ring Element

Let R be an associative unital algebra over a field k, let p be an element of R, and let R′ = R q | pqp = p . We obtain normal forms for elements of R′, and for elements of R′-modules arising by extension of scalars from R-modules. The details depend on where in the chain pR ∩ Rp ⊆ pR ∪ Rp ⊆ pR + Rp ⊆ R the unit 1 of R first appears. This investigation is motivated by a hoped-for application to the study of the possible forms of the monoid of isomorphism classes of finitely generated projective modules over a von Neumann regular ring; but that goal remains distant. We end with a normal form result for the algebra obtained by tying together a k-algebra R given with a nonzero element p satisfying 1 / ∈ pR+Rp and a k-algebra S given with a nonzero q satisfying 1 / ∈ qS+Sq, via the pair of relations p = pqp, q = qpq. 1. Motivation: monoids of projective modules It is known that the abelian monoid of isomorphism classes of finitely generated projective modules over a general ring is subject to no nonobvious restrictions – the obvious restrictions being (1) no two nonzero elements of the monoid have sum zero,