# On von Neumann regular elements in f-rings

On von Neumann regular elements in f-rings Let B be an Archimedean reduced f-ring. A positive element $${\omega}$$ ω in B is said to satisfy the property $${(\ast)}$$ ( * ) if for every f-ring A with identity e and every $${\ell}$$ ℓ -group homomorphism $${\gamma : A \rightarrow B}$$ γ : A → B with $${\gamma(e) = \omega}$$ γ ( e ) = ω , there exists a unique $${\ell}$$ ℓ -ring homomorphism $${\rho: B \rightarrow B}$$ ρ : B → B such that $${\gamma = \omega \rho}$$ γ = ω ρ and $${\rho(e)^{\perp \perp} = \omega^{\perp \perp}}$$ ρ ( e ) ⊥ ⊥ = ω ⊥ ⊥ . Boulabiar and Hager proved that any (positive) von Neumann regular element in B satisfies the property $${(\ast)}$$ ( * ) and proved that the converse holds in the C(X)-case. In this regard, they asked about this converse in the general case. Our main purpose in this note is to prove, via a counter-example, that the converse in question fails in general. In addition, we shall take the opportunity to extend the direct result obtained by Boulabiar and Hager, and to get the C(X)-case we were talking about in an easier way. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png algebra universalis Springer Journals

# On von Neumann regular elements in f-rings

, Volume 78 (1) – May 22, 2017
6 pages

/lp/springer_journal/on-von-neumann-regular-elements-in-f-rings-qXkdNTCiBv
Publisher
Springer International Publishing
Copyright © 2017 by Springer International Publishing
Subject
Mathematics; Algebra
ISSN
0002-5240
eISSN
1420-8911
D.O.I.
10.1007/s00012-017-0445-0
Publisher site
See Article on Publisher Site

### Abstract

Let B be an Archimedean reduced f-ring. A positive element $${\omega}$$ ω in B is said to satisfy the property $${(\ast)}$$ ( * ) if for every f-ring A with identity e and every $${\ell}$$ ℓ -group homomorphism $${\gamma : A \rightarrow B}$$ γ : A → B with $${\gamma(e) = \omega}$$ γ ( e ) = ω , there exists a unique $${\ell}$$ ℓ -ring homomorphism $${\rho: B \rightarrow B}$$ ρ : B → B such that $${\gamma = \omega \rho}$$ γ = ω ρ and $${\rho(e)^{\perp \perp} = \omega^{\perp \perp}}$$ ρ ( e ) ⊥ ⊥ = ω ⊥ ⊥ . Boulabiar and Hager proved that any (positive) von Neumann regular element in B satisfies the property $${(\ast)}$$ ( * ) and proved that the converse holds in the C(X)-case. In this regard, they asked about this converse in the general case. Our main purpose in this note is to prove, via a counter-example, that the converse in question fails in general. In addition, we shall take the opportunity to extend the direct result obtained by Boulabiar and Hager, and to get the C(X)-case we were talking about in an easier way.

### Journal

algebra universalisSpringer Journals

Published: May 22, 2017

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations