Artin Rings with Two-Generated Ideals

Rush, David E.

doi:10.4153/cmb-1992-019-3

Cited by 2 publications

(1 citation statement)

References 9 publications

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“…He also showed that a Noetherian ring R is an n-generated ring (for some n) if and only if there exists a positive integer m such that R M is an m-generated ring for each maximal ideal M of R [Gilmer 1972a;1972b]. For more on n-generated ideals and rings, we refer the readers to [Ameziane Hassani et al 1996;Cohen 1950;Gilmer 1972b;Heinzer and Lantz 1983;Matsuda 1979;McLean 1982;1993;Rush 1991;1992;Sally 1978;Shalev 1986].…”

Section: Introductionmentioning

confidence: 99%

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2016

Pacific J. Math.

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Let I be an ideal of the ring of Laurent polynomials K [x ±11 , . . . , x ±1 n ] with coefficients in a real-valued field (K, v). The fundamental theorem of tropical algebraic geometry states the equality trop(V (I)) = V (trop(I)) between the tropicalization trop(V (I)) of the closed subscheme V (I) ⊂ (K * ) n and the tropical variety V (trop(I)) associated to the tropicalization of the ideal trop(I).In this work we prove an analogous result for a differential ideal G of the ring of differential polynomials K [[t]]{x 1 , . . . , x n }, where K is an uncountable algebraically closed field of characteristic zero. We define the tropicalization trop(Sol(G)) of the set of solutions Sol(G) ⊂ K [[t]] n of G, and the set of solutions Sol(trop(G)) ⊂ P(‫ޚ‬ ≥0 ) n associated to the tropicalization of the ideal trop(G). These two sets are linked by a tropicalization morphism trop : Sol(G) → Sol(trop(G)).We show the equality trop(Sol(G)) = Sol(trop(G)), answering a question recently raised by D. Grigoriev.

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Section: Introductionmentioning

confidence: 99%

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2016

Pacific J. Math.

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A remark on the Noetherian property of power series rings

Kang

Toan

2016

Pacific J. Math.

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Let α be a (finite or infinite) cardinal number. An ideal of a ring R is called an α-generated ideal if it can be generated by a set with cardinality at most α. A ring R is called an α-generated ring if every ideal of R is an α-generated ideal. When α is finite, the class of α-generated rings has been studied in literature by scholars such as I. S. Cohen and R. Gilmer. In this paper, the class of α-generated rings when α is infinite (in particular, when α = ℵ 0 , the smallest infinite cardinal number) is considered. Surprisingly, it is proved that the concepts "ℵ 0 -generated ring" and "Noetherian ring" are the same for the power series ring R[[X]]. In other words, if every ideal of R[[X]] is countably generated, then each of them is in fact finitely generated. This shows a strange behavior of the power series ring R[[X]] compared to that of the polynomial ring R[X]. Indeed, for any infinite cardinal number α, it is proved that R is an α-generated ring if and only if R[X] is an α-generated ring, which is an analogue of the Hilbert basis theorem stating that R is a Noetherian ring if and only if R[X] is a Noetherian ring. Let ᏻ be the ring of algebraic integers. Under the continuum hypothesis, we show that ᏻ [[X]] contains an |ᏻ[[X]]|-generated (and hence uncountably generated) ideal which is not a β-generated ideal for any cardinal number β < |ᏻ [[X]]| and that the concepts "ℵ 1 -generated ring" and "ℵ 0 -generated ring" are different for the power series ring R [[X]].

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Artin Rings with Two-Generated Ideals

Cited by 2 publications

References 9 publications

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A remark on the Noetherian property of power series rings

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