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$f$-ring; OIRI-ring; strong order unit; $l$-ideal; nilpotent; annihilator; order ideal; ring ideal; unitable; archimedean
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.
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