Adjoining a universal inner inverse to a ring element

TL;DR

This paper develops normal forms for an algebra extension obtained by adjoining an inner inverse to a ring element, with potential applications to classifying projective modules over von Neumann regular rings.

Contribution

It introduces a method to derive normal forms for elements in an algebra extended by adjoining an inner inverse, depending on the position of the unit within certain submodules.

Findings

Normal forms for elements of the extended algebra are established.

The structure depends on the chain position of the unit in specific submodules.

Normal form results are also obtained for algebras tied via specific relations.

Abstract

Let $R$ be an associative unital algebra over a field $k,$ let $p$ be an element of $R,$ and let $R'=R\langle q\mid pqp= p\rangle.$ 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 $pR\cap Rp \subseteq pR\cup Rp \subseteq pR + Rp \subseteq R$ 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 $1\notin pR+Rp$ and a $k$-algebra $S$ given with a nonzero $q$ satisfying $1\notin qS+Sq,$ via the pair of relations $p=pqp,$ $q=qpq.$