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.
@article{116985,
author = {Melvin Henriksen and Suzanne Larson and Frank A. Smith},
title = {When is every order ideal a ring ideal?},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {32},
year = {1991},
pages = {411-416},
zbl = {0744.06008},
mrnumber = {1159787},
language = {en},
url = {http://dml.mathdoc.fr/item/116985}
}
Henriksen, Melvin; Larson, Suzanne; Smith, Frank A. When is every order ideal a ring ideal?. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) pp. 411-416. http://gdmltest.u-ga.fr/item/116985/
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