Adjoining a universal inner inverse to a ring element
Abstract
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 ⊂eq pR Rp ⊂eq pR + Rp ⊂eq 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.
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