# When is every order ideal a ring ideal?

Melvin Henriksen; Suzanne Larson; Frank A. Smith

Commentationes Mathematicae Universitatis Carolinae (1991)

- Volume: 32, Issue: 3, page 411-416
- ISSN: 0010-2628

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topHenriksen, 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 -

## References

top- Bigard A., Keimel K., Wolfenstein S., Groupes et Anneaux Réticulés, Lecture Notes in Mathematics 608, Springer-Verlag, New York, 1977. Zbl0384.06022MR0552653
- Basly M., Triki A., $F$-algebras in which order ideals are ring ideals, Proc. Konin. Neder. Akad. Wet. 91 (1988), 231-234. (1988) Zbl0662.46006MR0964828
- Feldman D., Henriksen M., $f$-rings, subdirect products of totally ordered rings, and the prime ideal theorem, ibid., 91 (1988), 121-126. (1988) Zbl0656.06017MR0952510
- Henriksen M., Isbell J., Lattice ordered rings and function rings, Pacific J. Math. 12 (1962), 533-565. (1962) Zbl0111.04302MR0153709
- Jech T., The Axiom of Choice, North Holland Publ. Co., Amsterdam, 1973. Zbl0259.02052MR0396271
- Luxemburg W., Zaanen A., Riesz Spaces, ibid., 1971. Zbl0231.46014

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