# Remarks on special ideals in lattices

Commentationes Mathematicae Universitatis Carolinae (1994)

- Volume: 35, Issue: 4, page 607-615
- ISSN: 0010-2628

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topBeran, Ladislav. "Remarks on special ideals in lattices." Commentationes Mathematicae Universitatis Carolinae 35.4 (1994): 607-615. <http://eudml.org/doc/247629>.

@article{Beran1994,

abstract = {The author studies some characteristic properties of semiprime ideals. The semiprimeness is also used to characterize distributive and modular lattices. Prime ideals are described as the meet-irreducible semiprime ideals. In relatively complemented lattices they are characterized as the maximal semiprime ideals. $D$-radicals of ideals are introduced and investigated. In particular, the prime radicals are determined by means of $\hat\{C\}$-radicals. In addition, a necessary and sufficient condition for the equality of prime radicals is obtained.},

author = {Beran, Ladislav},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {semiprime ideal; prime ideal; congruence of a lattice; allele; lattice polynomial; meet-irreducible element; kernel; forbidden exterior quotients; $D$-radical; prime radical; prime ideal; congruence; allele; lattice polynomial; kernel; forbidden exterior quotients; distributive lattices; semiprime ideals; modular lattices; -radicals; prime radicals},

language = {eng},

number = {4},

pages = {607-615},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {Remarks on special ideals in lattices},

url = {http://eudml.org/doc/247629},

volume = {35},

year = {1994},

}

TY - JOUR

AU - Beran, Ladislav

TI - Remarks on special ideals in lattices

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 1994

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 35

IS - 4

SP - 607

EP - 615

AB - The author studies some characteristic properties of semiprime ideals. The semiprimeness is also used to characterize distributive and modular lattices. Prime ideals are described as the meet-irreducible semiprime ideals. In relatively complemented lattices they are characterized as the maximal semiprime ideals. $D$-radicals of ideals are introduced and investigated. In particular, the prime radicals are determined by means of $\hat{C}$-radicals. In addition, a necessary and sufficient condition for the equality of prime radicals is obtained.

LA - eng

KW - semiprime ideal; prime ideal; congruence of a lattice; allele; lattice polynomial; meet-irreducible element; kernel; forbidden exterior quotients; $D$-radical; prime radical; prime ideal; congruence; allele; lattice polynomial; kernel; forbidden exterior quotients; distributive lattices; semiprime ideals; modular lattices; -radicals; prime radicals

UR - http://eudml.org/doc/247629

ER -

## References

top- Beran L., Orthomodular Lattices (Algebraic Approach), Reidel Dordrecht (1985). (1985) Zbl0558.06008MR0784029
- Beran L., Distributivity in finitely generated orthomodular lattices, Comment. Math. Univ. Carolinae 28 (1987), 433-435. (1987) Zbl0624.06008MR0912572
- Beran L., On semiprime ideals in lattices, J. Pure Appl. Algebra 64 (1990), 223-227. (1990) Zbl0703.06003MR1061299
- Beran L., On the rhomboidal heredity in ideal lattices, Comment. Math. Univ. Carolinae 33 (1992), 723-726. (1992) Zbl0782.06007MR1240194
- Birkhoff G., Lattice Theory, 3rd ed., American Math. Soc. Colloq. Publ., vol. XXV, Providence, 1967. Zbl0537.06001MR0227053
- Chevalier G., Semiprime ideals in orthomodular lattices, Comment. Math. Univ. Carolinae 29 (1988), 379-386. (1988) Zbl0655.06008MR0957406
- Dubreil-Jacotin M.L., Lesieur L., Croisot R., Leçons sur la théorie des treillis, des structures algébriques ordonnées et des treillis géometriques, Gauthier-Villars Paris (1953). (1953) Zbl0051.26005MR0057838
- Rav Y., Semiprime ideals in general lattices, J. Pure Appl. Algebra 56 (1989), 105-118. (1989) Zbl0665.06006MR0979666

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