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When is every order ideal a ring ideal? (English) Zbl 0744.06008

Summary: A lattice-ordered ring \(\mathbb{R}\) is called an OIRI-ring if each of its order ideals is a ring ideal. Generalizing earlier work of Basly and Triki, OIRI-rings are characterized as those \(f\)-rings \(\mathbb{R}\) such that \(\mathbb{R}/\mathbb{I}\) is contained in an \(f\)-ring with an identity element that is a strong order unit for some nil \(\ell\)-ideal \(\mathbb{I}\) of \(\mathbb{R}\). In particular, if \(P(\mathbb{R})\) denotes the set of nilpotent elements of the \(f\)-ring \(\mathbb{R}\), then \(\mathbb{R}\) is an OIRI-ring if and only if \(\mathbb{R}/P(\mathbb{R})\) is contained in an \(f\)-ring with an identity element that is a strong order unit.

MSC:

06F25 Ordered rings, algebras, modules
13C05 Structure, classification theorems for modules and ideals in commutative rings
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