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Irreducibility of ideals in a one-dimensional analytically irreducible ring
Valentina Barucci1; Faten Khouja2
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, Volume 2 (2010) no. 2, pp. 91-93.

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.

Published online:
DOI: 10.5802/acirm.40
Classification: 13A15, 13H10
Keywords: Numerical semigroup, canonical ideal, irreducible ideal.
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. 
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Valentina Barucci; Faten Khouja. Irreducibility of ideals in a one-dimensional analytically irreducible ring. Actes des rencontres du CIRM, Volume 2 (2010) no. 2, pp. 91-93. doi : 10.5802/acirm.40. https://acirm.centre-mersenne.org/articles/10.5802/acirm.40/

[1] V. Barucci, Decompositions of ideals into irreducible ideals in numerical semigroups. Journal of Commutative Algebra 2 (2010), 281-294. | DOI | MR | Zbl

[2] V. Barucci and R. Fröberg, One-dimensional almost Gorenstein rings. J. Algebra 188 (1997), 418-442. | DOI | MR | Zbl

[3] J. Jäger, Langenberechnung und kanonische Ideale in eindimensionalen Ringen. Arch. Math. 29 (1977), 504-512. | DOI | Zbl

[4] W. Vasconcelos, Computational Methods in Commutative Alegebra and Algebraic Geometry. Springer-Verlag, 1998. | DOI

[5] E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry. Birkhauser, 1984. | DOI

[6] E. Miller and B. Sturmfels, Combinatorial Commutative Algebra. Springer-Verlag, 2005. | DOI | Zbl

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