Primeness and semiprimeness in posets

Vilas S. Kharat; Khalid A. Mokbel

Mathematica Bohemica (2009)

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

Abstract

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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.

How to cite

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Kharat, 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

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  7. Halaš, R., Decompositions of directed sets with zero, Math. Slovaca 45 (1995), 9-17. (1995) MR1335835
  8. Halaš, R., Rachůnek, J., Polars and prime ideals in ordered sets, Discuss. Math., Algebra Stoch. Methods 15 (1995), 43-50. (1995) MR1369627
  9. 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
  10. 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

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