| Title:
|
When is every order ideal a ring ideal? (English) |
| Author:
|
Henriksen, M. |
| Author:
|
Larson, S. |
| Author:
|
Smith, F. A. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
32 |
| Issue:
|
3 |
| Year:
|
1991 |
| Pages:
|
411-416 |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
$f$-ring |
| Keyword:
|
OIRI-ring |
| Keyword:
|
strong order unit |
| Keyword:
|
$l$-ideal |
| Keyword:
|
nilpotent |
| Keyword:
|
annihilator |
| Keyword:
|
order ideal |
| Keyword:
|
ring ideal |
| Keyword:
|
unitable |
| Keyword:
|
archimedean |
| MSC:
|
06F25 |
| MSC:
|
13C05 |
| MSC:
|
16D15 |
| MSC:
|
16W80 |
| idZBL:
|
Zbl 0744.06008 |
| idMR:
|
MR1159787 |
| . |
| Date available:
|
2009-01-08T17:45:30Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/116985 |
| . |
| Reference:
|
[BKW] Bigard A., Keimel K., Wolfenstein S.: Groupes et Anneaux Réticulés.Lecture Notes in Mathematics 608, Springer-Verlag, New York, 1977. Zbl 0384.06022, MR 0552653 |
| Reference:
|
[BT] Basly M., Triki A.: $F$-algebras in which order ideals are ring ideals.Proc. Konin. Neder. Akad. Wet. 91 (1988), 231-234. Zbl 0662.46006, MR 0964828 |
| Reference:
|
[FH] Feldman D., Henriksen M.: $f$-rings, subdirect products of totally ordered rings, and the prime ideal theorem.ibid., 91 (1988), 121-126. Zbl 0656.06017, MR 0952510 |
| Reference:
|
[HI] Henriksen M., Isbell J.: Lattice ordered rings and function rings.Pacific J. Math. 12 (1962), 533-565. Zbl 0111.04302, MR 0153709 |
| Reference:
|
[J] Jech T.: The Axiom of Choice.North Holland Publ. Co., Amsterdam, 1973. Zbl 0259.02052, MR 0396271 |
| Reference:
|
[LZ] Luxemburg W., Zaanen A.: Riesz Spaces.ibid., 1971. Zbl 0231.46014 |
| . |
