Algebra Univers. 78 (2017) 119–124
Published online May 22, 2017
© Springer International Publishing 2017
On von Neumann regular elements in f-rings
Youssef Azouzi and Mohamed Amine Ben Amor
Abstract. Let B be an Archimedean reduced f-ring. A positive element w in B is
said to satisfy the property (∗) if for every f-ring A with identity e and every -group
homomorphism γ : A → B with γ(e)=w, there exists a unique -ring homomorphism
ρ: B → B such that γ = wρ and ρ(e)
. Boulabiar and Hager proved
that any (positive) von Neumann regular element in B satisﬁes the property (∗) 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.
There is extensive literature on the connections between - homomorphisms
and ring homomorphisms on f-rings (see, for instance, [2,5,8,9]). Amongst
other results, Boulabiar and Hager proved the following (see [6, Theorem 3.3]).
Theorem 1.1. Let A and B be Archimedean reduced f-rings, and suppose that
A has an identity e. If w is a von Neumann regular element in B, then for
any -homomorphism γ : A → B with w = γ(e), there exists a unique -ring
homomorphism ρ: A → B such that
γ = wρ and w
In this work, we provide a generalization of Theorem 1.1. Indeed, we will
neither assume that B is reduced nor that A is Archimedean. On the other
hand, Boulabiar and Hager asked about the converse problem. They proved
that it holds for the C(X)-case. However, we will prove via a counter example
that the converse problem fails to be true in the general case. Also, we shall
give an alternative proof of the C(X)-case which we believe simpler and more
straightforward than the proof given by Boulabiar and Hager. Here, C(X)
denotes the Archimedean f -ring of all real-valued continuous functions on a
completely regular space X.
We will start our study with a preliminary section in which we recall some
facts about homomorphisms on f -rings and -groups.
Presented by W. McGovern.
Received October 4, 2015; accepted in ﬁnal form January 19, 2017.
2010 Mathematics Subject Classiﬁcation: Primary: 06F25; Secondary: 06F15, 46E05,
Key words and phrases: -group, f-ring, -homomorphism, von Neumann regular, ring
of continuous functions.