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Keywords:
$f$-ring; OIRI-ring; strong order unit; $l$-ideal; nilpotent; annihilator; order ideal; ring ideal; unitable; archimedean
$f$-ring; OIRI-ring; strong order unit; $l$-ideal; nilpotent; annihilator; order ideal; ring ideal; unitable; archimedean
Summary:
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.
References:
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[BT] Basly M., Triki A.: $F$-algebras in which order ideals are ring ideals. Proc. Konin. Neder. Akad. Wet. 91 (1988), 231-234. MR 0964828 | Zbl 0662.46006
[FH] Feldman D., Henriksen M.: $f$-rings, subdirect products of totally ordered rings, and the prime ideal theorem. ibid., 91 (1988), 121-126. MR 0952510 | Zbl 0656.06017
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