Open Access
2018 Strong inner inverses in endomorphism rings of vector spaces
George M. Bergman
Publ. Mat. 62(1): 253-284 (2018). DOI: 10.5565/PUBLMAT6211812


For $V$ a vector space over a field, or more generally, over a division ring, it is well-known that every $x\in\operatorname{End}(V)$ has an inner inverse; that is, that there exists $y\in\operatorname{End}(V)$ satisfying $xyx=x$. We show here that a large class of such $x$ have inner inverses $y$ that satisfy with $x$ an infinite family of additional monoid relations, making the monoid generated by $x$ and $y$ what is known as an inverse monoid (definition recalled). We obtain consequences of these relations, and related results.

P. Nielsen and J. Šter [16] show that a much larger class of elements $x$ of rings $R$, including all elements of von Neumann regular rings, have inner inverses satisfying arbitrarily large finite subsets of the abovementioned set of relations. But we show by example that the endomorphism ring of any infinite-dimensional vector space contains elements having no inner inverse that simultaneously satisfies all those relations.

A tangential result gives a condition on an endomap $x$ of a set $S$ that is necessary and sufficient for $x$ to have a strong inner inverse in the monoid of all endomaps of $S$.


Download Citation

George M. Bergman. "Strong inner inverses in endomorphism rings of vector spaces." Publ. Mat. 62 (1) 253 - 284, 2018.


Received: 14 November 2016; Revised: 20 February 2017; Published: 2018
First available in Project Euclid: 16 December 2017

zbMATH: 1381.16029
MathSciNet: MR3738191
Digital Object Identifier: 10.5565/PUBLMAT6211812

Primary: 16E50 , 16S50 , 16U99 , 20M18
Secondary: 16S15 , 16S36

Keywords: endomorphism ring of a vector space , inner inverse to a ring element , inverse monoid

Rights: Copyright © 2018 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.62 • No. 1 • 2018
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