When is every order ideal a ring ideal?

Henriksen, Melvin, Larson, Suzanne, and Smith, Frank A.. "When is every order ideal a ring ideal?." Commentationes Mathematicae Universitatis Carolinae 32.3 (1991): 411-416. <http://eudml.org/doc/247286>.

@article{Henriksen1991,
abstract = {A lattice-ordered ring $\mathbb \{R\}$ is called an 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 $\mathbb \{R\}$ such that $\mathbb \{R\}/\mathbb \{I\}$ is contained in an $f$-ring with an identity element that is a strong order unit for some nil $l$-ideal $\mathbb \{I\}$ of $\mathbb \{R\}$. In particular, if $P(\mathbb \{R\})$ denotes the set of nilpotent elements of the $f$-ring $\mathbb \{R\}$, then $\mathbb \{R\}$ is an OIRI-ring if and only if $\mathbb \{R\}/P(\mathbb \{R\})$ is contained in an $f$-ring with an identity element that is a strong order unit.},
author = {Henriksen, Melvin, Larson, Suzanne, Smith, Frank A.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$f$-ring; OIRI-ring; strong order unit; $l$-ideal; nilpotent; annihilator; order ideal; ring ideal; unitable; archimedean; lattice-ordered ring; OIRI-ring; order ideals; ring ideal; -ideal; nilpotent; -ring; strong order unit},
language = {eng},
number = {3},
pages = {411-416},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {When is every order ideal a ring ideal?},
url = {http://eudml.org/doc/247286},
volume = {32},
year = {1991},
}

TY - JOUR
AU - Henriksen, Melvin
AU - Larson, Suzanne
AU - Smith, Frank A.
TI - When is every order ideal a ring ideal?
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 3
SP - 411
EP - 416
AB - A lattice-ordered ring $\mathbb {R}$ is called an 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 $\mathbb {R}$ such that $\mathbb {R}/\mathbb {I}$ is contained in an $f$-ring with an identity element that is a strong order unit for some nil $l$-ideal $\mathbb {I}$ of $\mathbb {R}$. In particular, if $P(\mathbb {R})$ denotes the set of nilpotent elements of the $f$-ring $\mathbb {R}$, then $\mathbb {R}$ is an OIRI-ring if and only if $\mathbb {R}/P(\mathbb {R})$ is contained in an $f$-ring with an identity element that is a strong order unit.
LA - eng
KW - $f$-ring; OIRI-ring; strong order unit; $l$-ideal; nilpotent; annihilator; order ideal; ring ideal; unitable; archimedean; lattice-ordered ring; OIRI-ring; order ideals; ring ideal; -ideal; nilpotent; -ring; strong order unit
UR - http://eudml.org/doc/247286
ER -