# Primeness and semiprimeness in posets

Vilas S. Kharat; Khalid A. Mokbel

Mathematica Bohemica (2009)

- Volume: 134, Issue: 1, page 19-30
- ISSN: 0862-7959

## Access Full Article

top## Abstract

top## How to cite

topKharat, Vilas S., and Mokbel, Khalid A.. "Primeness and semiprimeness in posets." Mathematica Bohemica 134.1 (2009): 19-30. <http://eudml.org/doc/38069>.

@article{Kharat2009,

abstract = {The concept of a semiprime ideal in a poset is introduced. Characterizations of semiprime ideals in a poset $P$ as well as characterizations of a semiprime ideal to be prime in $P$ are obtained in terms of meet-irreducible elements of the lattice of ideals of $P$ and in terms of maximality of ideals. Also, prime ideals in a poset are characterized.},

author = {Kharat, Vilas S., Mokbel, Khalid A.},

journal = {Mathematica Bohemica},

keywords = {semiprime ideal; prime ideal; meet-irreducible element; $I$-atom; semiprime ideal; prime ideal; meet-irreducible element; -atom},

language = {eng},

number = {1},

pages = {19-30},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {Primeness and semiprimeness in posets},

url = {http://eudml.org/doc/38069},

volume = {134},

year = {2009},

}

TY - JOUR

AU - Kharat, Vilas S.

AU - Mokbel, Khalid A.

TI - Primeness and semiprimeness in posets

JO - Mathematica Bohemica

PY - 2009

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 134

IS - 1

SP - 19

EP - 30

AB - The concept of a semiprime ideal in a poset is introduced. Characterizations of semiprime ideals in a poset $P$ as well as characterizations of a semiprime ideal to be prime in $P$ are obtained in terms of meet-irreducible elements of the lattice of ideals of $P$ and in terms of maximality of ideals. Also, prime ideals in a poset are characterized.

LA - eng

KW - semiprime ideal; prime ideal; meet-irreducible element; $I$-atom; semiprime ideal; prime ideal; meet-irreducible element; -atom

UR - http://eudml.org/doc/38069

ER -

## References

top- Beran, L., 10.1016/0022-4049(90)90058-P, J. Pure Appl. Algebra 64 (1990), 223-227. (1990) Zbl0703.06003MR1061299DOI10.1016/0022-4049(90)90058-P
- Beran, L., Remarks on special ideals in lattices, Comment. Math. Univ. Carol. 35 (1994), 607-615. (1994) Zbl0812.06002MR1321231
- Beran, L., Length of ideals in lattices, Collect. Math. 46 (1995), 21-33. (1995) Zbl0842.06006MR1366126
- Beran, L., Some types of implicative ideals, Comment. Math. Univ. Carol. 39 (1998), 219-225. (1998) Zbl0938.06004MR1651930
- Grätzer, G., General Lattice Theory, Birkhäuser, second edition (1998). (1998) MR1670580
- Halaš, R., Characterization of distributive sets by generalized annihilators, Arch. Math. (Brno) 30 (1994), 25-27. (1994) MR1282110
- Halaš, R., Decompositions of directed sets with zero, Math. Slovaca 45 (1995), 9-17. (1995) MR1335835
- Halaš, R., Rachůnek, J., Polars and prime ideals in ordered sets, Discuss. Math., Algebra Stoch. Methods 15 (1995), 43-50. (1995) MR1369627
- Larmerová, J., Rachůnek, J., Translations of distributive and modular ordered sets, Acta Univ. Palack. Olomouc, Fac. Rer. Nat. 91, Math. 27 (1988), 13-23. (1988) MR1039879
- Rav, Y., 10.1016/0022-4049(89)90140-0, J. Pure Appl. Algebra 56 (1989), 105-118. (1989) Zbl0665.06006MR0979666DOI10.1016/0022-4049(89)90140-0

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.