Irreducibility of ideals in a one-dimensional analytically irreducible ring. (English) Zbl 1434.13005

Summary: 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 E. Kunz [Introduction to commutative algebra and algebraic geometry. Boston-Basel-Stuttgart: Birkhäuser (1985; Zbl 0563.13001)], W. V. Vasconcelos [Computational methods of commutative algebra and algebraic geometry. Berlin: Springer (1998; Zbl 0896.13021)] and J. Jäger [Arch. Math. 29, 504–512 (1977; Zbl 0374.13006)] to obtain this result, which is new. We finally give an algorithm to compute the components of an irredundant decomposition of a nonzero ideal.


13A15 Ideals and multiplicative ideal theory in commutative rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
Full Text: DOI Link


[1] V. Barucci, \(Decompositions of ideals into irreducible ideals in numerical semigroups\). Journal of Commutative Algebra 2 (2010), 281-294. · Zbl 1237.20056
[2] V. Barucci and R. Fröberg, \(One-dimensional almost Gorenstein rings\). J. Algebra 188 (1997), 418-442. · Zbl 0874.13018
[3] J. Jäger, \(Langenberechnung und kanonische Ideale in eindimensionalen Ringen\). Arch. Math. 29 (1977), 504-512. · Zbl 0374.13006
[4] W. Vasconcelos, \(Computational Methods in Commutative Alegebra and Algebraic Geometry\). Springer-Verlag, 1998. · Zbl 0896.13021
[5] E. Kunz, \(Introduction to Commutative Algebra and Algebraic Geometry\). Birkhauser, 1984.
[6] E. Miller and B. Sturmfels, \(Combinatorial Commutative Algebra\). Springer-Verlag, 2005. | Copyright Cellule MathDoc 2018 · Zbl 1090.13001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.