Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T23:18:31.638Z Has data issue: false hasContentIssue false
  • English
  • Français

Irreducible decomposition of binomial ideals

Part of: Arithmetic rings and other special rings Computational aspects and applications General commutative ring theory Theory of modules and ideals Semigroups Algebraic combinatorics

Published online by Cambridge University Press:  01 April 2016

Thomas Kahle ,
Ezra Miller  and
Christopher O’Neill
Thomas Kahle
Affiliation:
Faculty of Mathematics, Otto-von-Guericke Universität, Magdeburg, Germany email [email protected]
Ezra Miller
Affiliation:
Mathematics Department, Duke University, Durham, NC 27708, USA email [email protected]
Christopher O’Neill
Affiliation:
Mathematics Department, Texas A&M University, TX 77843, USA email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Building on coprincipal mesoprimary decomposition [Kahle and Miller, Decompositions of commutative monoid congruences and binomial ideals, Algebra and Number Theory 8 (2014), 1297–1364], we combinatorially construct an irreducible decomposition of any given binomial ideal. In a parallel manner, for congruences in commutative monoids we construct decompositions that are direct combinatorial analogues of binomial irreducible decompositions, and for binomial ideals we construct decompositions into ideals that are as irreducible as possible while remaining binomial. We provide an example of a binomial ideal that is not an intersection of irreducible binomial ideals, thus answering a question of Eisenbud and Sturmfels [Binomial ideals, Duke Math. J. 84 (1996), 1–45].

Keywords

binomial idealirreducible idealmonoid congruenceprimary decompositionirreducible decompositionmesoprimary decomposition

MSC classification

Primary: 13C05: Structure, classification theorems 05E40: Combinatorial aspects of commutative algebra 20M25: Semigroup rings, multiplicative semigroups of rings
Secondary: 05E15: Combinatorial aspects of groups and algebras 13F99: None of the above, but in this section 13A02: Graded rings 13P99: None of the above, but in this section
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 6 , June 2016 , pp. 1319 - 1332
Copyright
© The Authors 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bass, H., Injective dimension in Noetherian rings, Trans. Amer. Math. Soc. 102 (1962), 18–29.Google Scholar
Eisenbud, D. and Sturmfels, B., Binomial ideals, Duke Math. J. 84 (1996), 1–45.Google Scholar
Kahle, T. and Miller, E., Decompositions of commutative monoid congruences and binomial ideals, Algebra and Number Theory 8 (2014), 1297–1364.CrossRefGoogle Scholar
Miller, E., Cohen–Macaulay quotients of normal semigroup rings via irreducible resolutions, Math. Res. Lett. 9 (2002), 117–128.CrossRefGoogle Scholar
Miller, E. and Sturmfels, B., Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227 (Springer, New York, NY, 2005).Google Scholar
Vasconcelos, W. V., Computational methods in commutative algebra and algebraic geometry, Algorithms and Computation in Mathematics, vol. 2 (Springer, Berlin, 1998).Google Scholar