Ideals of polynomial semirings in tropical mathematics

Zur Izhakian; Louis Rowen

doi:10.1142/S0219498812501435

Ideals of polynomial semirings in tropical mathematics

Zur Izhakian, Louis Rowen

Department of Mathematics

Bar-Ilan University

Research output: Contribution to journal › Article › peer-review

Abstract

We describe the ideals, especially the prime ideals, of semirings of polynomials over layered domains, and in particular over supertropical domains. Since there are so many of them, special attention is paid to the ideals arising from layered varieties, for which we prove that every prime ideal is a consequence of finitely many binomials. We also obtain layered tropical versions of the classical Principal Ideal Theorem and Hilbert Basis Theorem.

Original language	English
Article number	1250143
Journal	Journal of Algebra and its Applications
Volume	12
Issue number	2
DOIs	https://doi.org/10.1142/S0219498812501435
State	Published - Mar 2013

Keywords

Hilbert Basis Theorem
Noetherian ideal theory
Principal Ideal Theorem
Tropical and supertropical algebra
layered algebra
layered ideals

All Science Journal Classification (ASJC) codes

Algebra and Number Theory
Applied Mathematics

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10.1142/S0219498812501435

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title = "Ideals of polynomial semirings in tropical mathematics",

abstract = "We describe the ideals, especially the prime ideals, of semirings of polynomials over layered domains, and in particular over supertropical domains. Since there are so many of them, special attention is paid to the ideals arising from layered varieties, for which we prove that every prime ideal is a consequence of finitely many binomials. We also obtain layered tropical versions of the classical Principal Ideal Theorem and Hilbert Basis Theorem.",

keywords = "Hilbert Basis Theorem, Noetherian ideal theory, Principal Ideal Theorem, Tropical and supertropical algebra, layered algebra, layered ideals",

author = "Zur Izhakian and Louis Rowen",

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AU - Rowen, Louis

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Y1 - 2013/3

N2 - We describe the ideals, especially the prime ideals, of semirings of polynomials over layered domains, and in particular over supertropical domains. Since there are so many of them, special attention is paid to the ideals arising from layered varieties, for which we prove that every prime ideal is a consequence of finitely many binomials. We also obtain layered tropical versions of the classical Principal Ideal Theorem and Hilbert Basis Theorem.

AB - We describe the ideals, especially the prime ideals, of semirings of polynomials over layered domains, and in particular over supertropical domains. Since there are so many of them, special attention is paid to the ideals arising from layered varieties, for which we prove that every prime ideal is a consequence of finitely many binomials. We also obtain layered tropical versions of the classical Principal Ideal Theorem and Hilbert Basis Theorem.

KW - Hilbert Basis Theorem

KW - Noetherian ideal theory

KW - Principal Ideal Theorem

KW - Tropical and supertropical algebra

KW - layered algebra

KW - layered ideals

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DO - 10.1142/S0219498812501435

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SN - 0219-4988

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JO - Journal of Algebra and its Applications

JF - Journal of Algebra and its Applications

IS - 2

M1 - 1250143

ER -

Ideals of polynomial semirings in tropical mathematics

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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