M. Henriksen, S. Larson, F.A. Smith
When is every order ideal a ring ideal?

Comment.Math.Univ.Carolinae 32,3 (1991) 411-416.

Abstract:A lattice-ordered ring $\Bbb R$ 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 $\Bbb R$ such that $\Bbb R/\Bbb I$ is contained in an $f$-ring with an identity element that is a strong order unit for some nil $l$-ideal $\Bbb I$ of $\Bbb R$. In particular, if $P(\Bbb R)$ denotes the set of nilpotent elements of the $f$-ring $\Bbb R$, then $\Bbb R$ is an OIRI-ring if and only if $\Bbb R/P(\Bbb R)$ is contained in an $f$-ring with an identity element that is a strong order unit.

Keywords: $f$-ring, OIRI-ring, strong order unit, $l$-ideal, nilpotent, annihilator, order ideal, ring ideal, unitable, archimedean
AMS Subject Classification: 06F25, 13C05

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