Invertible ideals and Gaussian semirings

Shaban Ghalandarzadeh; Peyman Nasehpour; Rafieh Razavi

Archivum Mathematicum (2017)

  • Volume: 053, Issue: 3, page 179-192
  • ISSN: 0044-8753

Abstract

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In the first section, we introduce the notions of fractional and invertible ideals of semirings and characterize invertible ideals of a semidomain. In section two, we define Prüfer semirings and characterize them in terms of valuation semirings. In this section, we also characterize Prüfer semirings in terms of some identities over its ideals such as ( I + J ) ( I ∩ J ) = I J for all ideals I , J of S . In the third section, we give a semiring version for the Gilmer-Tsang Theorem, which states that for a suitable family of semirings, the concepts of Prüfer and Gaussian semirings are equivalent. At last, we end this paper by giving a plenty of examples for proper Gaussian and Prüfer semirings.

How to cite

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Ghalandarzadeh, Shaban, Nasehpour, Peyman, and Razavi, Rafieh. "Invertible ideals and Gaussian semirings." Archivum Mathematicum 053.3 (2017): 179-192. <http://eudml.org/doc/294113>.

@article{Ghalandarzadeh2017,
abstract = {In the first section, we introduce the notions of fractional and invertible ideals of semirings and characterize invertible ideals of a semidomain. In section two, we define Prüfer semirings and characterize them in terms of valuation semirings. In this section, we also characterize Prüfer semirings in terms of some identities over its ideals such as $(I + J)(I \cap J) = IJ$ for all ideals $I$, $J$ of $S$. In the third section, we give a semiring version for the Gilmer-Tsang Theorem, which states that for a suitable family of semirings, the concepts of Prüfer and Gaussian semirings are equivalent. At last, we end this paper by giving a plenty of examples for proper Gaussian and Prüfer semirings.},
author = {Ghalandarzadeh, Shaban, Nasehpour, Peyman, Razavi, Rafieh},
journal = {Archivum Mathematicum},
keywords = {semiring; semiring polynomials; Gaussian semiring; cancellation ideals; invertible ideals},
language = {eng},
number = {3},
pages = {179-192},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Invertible ideals and Gaussian semirings},
url = {http://eudml.org/doc/294113},
volume = {053},
year = {2017},
}

TY - JOUR
AU - Ghalandarzadeh, Shaban
AU - Nasehpour, Peyman
AU - Razavi, Rafieh
TI - Invertible ideals and Gaussian semirings
JO - Archivum Mathematicum
PY - 2017
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 053
IS - 3
SP - 179
EP - 192
AB - In the first section, we introduce the notions of fractional and invertible ideals of semirings and characterize invertible ideals of a semidomain. In section two, we define Prüfer semirings and characterize them in terms of valuation semirings. In this section, we also characterize Prüfer semirings in terms of some identities over its ideals such as $(I + J)(I \cap J) = IJ$ for all ideals $I$, $J$ of $S$. In the third section, we give a semiring version for the Gilmer-Tsang Theorem, which states that for a suitable family of semirings, the concepts of Prüfer and Gaussian semirings are equivalent. At last, we end this paper by giving a plenty of examples for proper Gaussian and Prüfer semirings.
LA - eng
KW - semiring; semiring polynomials; Gaussian semiring; cancellation ideals; invertible ideals
UR - http://eudml.org/doc/294113
ER -

References

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