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

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Publisher
Springer International Publishing
Copyright
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

References

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