Geodesic
When is every order ideal a ring ideal?
Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 3, pp. 411-416.

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A lattice-ordered ring $\Bbb R$ is called an {\sl 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.
Classification : 06F25, 13C05, 16D15, 16W80
Keywords: $f$-ring; OIRI-ring; strong order unit; $l$-ideal; nilpotent; annihilator; order ideal; ring ideal; unitable; archimedean 
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Henriksen, M.; Larson, S.; Smith, F. A. When is every order ideal a ring ideal?. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 3, pp. 411-416. http://geodesic.mathdoc.fr/item/CMUC_1991__32_3_a0/