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# Article

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Keywords:
\$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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