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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⊆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.

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