# Irreducibility of ideals in a one-dimensional analytically irreducible ring

• [1] Dipartimento di Matematica, Sapienza, Università di Roma, Piazzale A. Moro 2, 00185 Rome, ITALY
• [2] Department of Mathematics, Faculty of Sciences, 5000 Monastir, TUNISIA.
• Volume: 2, Issue: 2, page 91-93
• ISSN: 2105-0597

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## Abstract

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Let $R$ be a one-dimensional analytically irreducible ring and let $I$ be an integral ideal of $R$. We study the relation between the irreducibility of the ideal $I$ in $R$ and the irreducibility of the corresponding semigroup ideal $v\left(I\right)$. It turns out that if $v\left(I\right)$ is irreducible, then $I$ is irreducible, but the converse does not hold in general. We collect some known results taken from [5], [4], [3] to obtain this result, which is new. We finally give an algorithm to compute the components of an irredundant decomposition of a nonzero ideal.

## How to cite

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Barucci, Valentina, and Khouja, Faten. "Irreducibility of ideals in a one-dimensional analytically irreducible ring." Actes des rencontres du CIRM 2.2 (2010): 91-93. <http://eudml.org/doc/196294>.

@article{Barucci2010,
abstract = {Let $R$ be a one-dimensional analytically irreducible ring and let $I$ be an integral ideal of $R$. We study the relation between the irreducibility of the ideal $I$ in $R$ and the irreducibility of the corresponding semigroup ideal $v(I)$. It turns out that if $v(I)$ is irreducible, then $I$ is irreducible, but the converse does not hold in general. We collect some known results taken from [5], [4], [3] to obtain this result, which is new. We finally give an algorithm to compute the components of an irredundant decomposition of a nonzero ideal.},
affiliation = {Dipartimento di Matematica, Sapienza, Università di Roma, Piazzale A. Moro 2, 00185 Rome, ITALY; Department of Mathematics, Faculty of Sciences, 5000 Monastir, TUNISIA.},
author = {Barucci, Valentina, Khouja, Faten},
journal = {Actes des rencontres du CIRM},
keywords = {Numerical semigroup; canonical ideal; irreducible ideal; numerical semigroup; class semigroup; reduction number},
language = {eng},
number = {2},
pages = {91-93},
publisher = {CIRM},
title = {Irreducibility of ideals in a one-dimensional analytically irreducible ring},
url = {http://eudml.org/doc/196294},
volume = {2},
year = {2010},
}

TY - JOUR
AU - Barucci, Valentina
AU - Khouja, Faten
TI - Irreducibility of ideals in a one-dimensional analytically irreducible ring
JO - Actes des rencontres du CIRM
PY - 2010
PB - CIRM
VL - 2
IS - 2
SP - 91
EP - 93
AB - Let $R$ be a one-dimensional analytically irreducible ring and let $I$ be an integral ideal of $R$. We study the relation between the irreducibility of the ideal $I$ in $R$ and the irreducibility of the corresponding semigroup ideal $v(I)$. It turns out that if $v(I)$ is irreducible, then $I$ is irreducible, but the converse does not hold in general. We collect some known results taken from [5], [4], [3] to obtain this result, which is new. We finally give an algorithm to compute the components of an irredundant decomposition of a nonzero ideal.
LA - eng
KW - Numerical semigroup; canonical ideal; irreducible ideal; numerical semigroup; class semigroup; reduction number
UR - http://eudml.org/doc/196294
ER -

## References

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1. V. Barucci, Decompositions of ideals into irreducible ideals in numerical semigroups. Journal of Commutative Algebra 2 (2010), 281-294. Zbl1237.20056MR2728145
2. V. Barucci and R. Fröberg, One-dimensional almost Gorenstein rings. J. Algebra 188 (1997), 418-442. Zbl0874.13018MR1435367
3. J. Jäger, Langenberechnung und kanonische Ideale in eindimensionalen Ringen. Arch. Math. 29 (1977), 504-512. Zbl0374.13006MR463156
4. W. Vasconcelos, Computational Methods in Commutative Alegebra and Algebraic Geometry. Springer-Verlag, 1998. Zbl0896.13021MR1484973
5. E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry. Birkhauser, 1984. MR789602
6. E. Miller and B. Sturmfels, Combinatorial Commutative Algebra. Springer-Verlag, 2005. Zbl1090.13001MR2110098

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