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

Valentina Barucci^{[1]}; Faten Khouja^{[2]}

- [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.

Actes des rencontres du CIRM (2010)

- Volume: 2, Issue: 2, page 91-93
- ISSN: 2105-0597

## Access Full Article

top## Abstract

top## How to cite

topBarucci, 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

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

## NotesEmbed ?

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