A geometric characterization of ring homomorphisms on \(f\)-rings (Q2847686)
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scientific article; zbMATH DE number 6207500
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A geometric characterization of ring homomorphisms on \(f\)-rings |
scientific article; zbMATH DE number 6207500 |
Statements
11 September 2013
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basic element
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convex set
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extreme point
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\(f\)-ring
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\(\ell\)-group
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ring homomorphism
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A geometric characterization of ring homomorphisms on \(f\)-rings (English)
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Let \(A\) be an \(f\)-ring with identity and \(B\) an Archimedean \(f\)-ring. The authors consider group homomorphisms from \(A\) to \(B\) that map the identity to a fixed idempotent element \(w\) in \(B\). They show that such a mapping is a ring homomorphism if and only if it is an extreme point of the set of all such maps. They derive a characterization of such ring homomorphisms in terms of a Gelfand-type transform. They finally show that in case \(u=1\), the homomorphisms are (up to multiplicative constants) the basis elements of the \(\ell\)-group of all bounded group homomorphisms from \(A\) to \(\mathbb R\).
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