Adjoining a universal inner inverse to a ring element

George M. Bergman

doi:10.1016/j.jalgebra.2015.11.008

Adjoining a universal inner inverse to a ring element

Rings and Algebras 2015-11-23 v2

Authors: George M. Bergman

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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 $p=pqp,$ $q=qpq.$

Keywords

commutative ring theory module theory commutative algebra

Cite

@article{arxiv.1505.02312,
  title  = {Adjoining a universal inner inverse to a ring element},
  author = {George M. Bergman},
  journal= {arXiv preprint arXiv:1505.02312},
  year   = {2015}
}

Comments

28 pages. Results on mutual inner inverses added at end of earlier version, and much clarification of wording etc.. After publication, any updates, errata, related references etc. found will be recorded at http://math.berkeley.edu/~gbergman/papers

Adjoining a universal inner inverse to a ring element

Abstract

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