Coherence of polynomial rings and bounds in polynomial ideals

Sabbagh, Gabriel

doi:10.1016/0021-8693(74)90128-8

Cited by 15 publications

(7 citation statements)

References 12 publications

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“…Soublin stated without proof in [4], and again in [5], that the polynomial ring R[X] is coherent. Recently, Sabbagh proved that a ring of polynomials in any number of indeterminates over R is coherent [3]. Carson obtained similar results for certain noncommutative rings [1].…”

Section: The Ring Of Polynomials Over a Von Neumann Regular Ring P Jsupporting

confidence: 50%

The ring of polynomial over a von Neumann regular ring

McCarthy¹

1973

Proc. Amer. Math. Soc.

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Abstract.It is shown that the ring of polynomials in one indeterminate over a commutative von Neumann regular ring with identity element is semihereditary.Let R be a commutative von Neumann regular ring with identity element. Soublin stated without proof in [4], and again in [5] Suppose «>0 and that the inclusion we are asserting holds whenf(X) is replaced by a polynomial of degree less than «.

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Section: The Ring Of Polynomials Over a Von Neumann Regular Ring P Jsupporting

confidence: 50%

The ring of polynomial over a von Neumann regular ring

McCarthy¹

1973

Proc. Amer. Math. Soc.

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“…It follows from the principle of infinite finite presentation [4, (11.3.9.1)] that / is finitely presented. This result has also been proved in [9]-(b) A is semihereditary. If fi = ^4[x,, .…”

supporting

confidence: 59%

Coherence of Polynomial Rings

Greenberg¹,

Vasconcelos²

1976

Proceedings of the American Mathematical Society

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“…By the remark above, the previous lemma and example (1), this implies that the class of von Neumann regular rings is extremely coherent with the same uniformity functions as in (1). (This was first observed by Sabbagh [30].) (3) The class of DVRs is extremely coherent, by the proof of Theorem 4.1 in…”

Section: Examplesmentioning

confidence: 73%

“…There exist coherent rings R which are not stably coherent [35]. We have the following theorem proved by Vasconcelos, after a conjecture by Sabbagh ( [30], p. 502). For an efficient proof based on work of Alfonsi see [18], Chapter 7.…”

Section: Homogeneous Linear Equations In Polynomial Ringsmentioning

confidence: 97%

“…We prove Theorem B in Section 4, using [25] for the implication (1) ⇒ (2) and by combining a result of Sabbagh [30] with an elementary lemma of Cherlin [10] for the converse. Note that condition (1) in Theorem B is satisfied if R is an Artinian local ring, yielding a result on uniform bounds for the ideal membership problem over R originally proved by Schoutens [31].…”

Section: Introductionmentioning

confidence: 99%

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Bounds and definability in polynomial rings

Aschenbrenner

2005

The Quarterly Journal of Mathematics

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Abstract. We study questions around the existence of bounds and the dependence on parameters for linear-algebraic problems in polynomial rings over rings of an arithmetic flavor. In particular, we show that the module of syzygies of polynomials f 1 , . . . , fn ∈ R[X 1 , . . . , X N ] with coefficients in a Prüfer domain R can be generated by elements whose degrees are bounded by a number only depending on N , n and the degree of the f j . This implies that if R is a Bézout domain, then the generators can be parametrized in terms of the coefficients of f 1 , . . . , fn using the ring operations and a certain division function, uniformly in R.

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Coherence of polynomial rings and bounds in polynomial ideals

Cited by 15 publications

References 12 publications

The ring of polynomial over a von Neumann regular ring

The ring of polynomial over a von Neumann regular ring

Coherence of Polynomial Rings

Bounds and definability in polynomial rings

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