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On strongly affine extensions of commutative rings

Nabil Zeidi (2020)

Czechoslovak Mathematical Journal

A ring extension R ⊆ S is said to be strongly affine if each R -subalgebra of S is a finite-type R -algebra. In this paper, several characterizations of strongly affine extensions are given. For instance, we establish that if R is a quasi-local ring of finite dimension, then R ⊆ S is integrally closed and strongly affine if and only if R ⊆ S is a Prüfer extension (i.e. ( R , S ) is a normal pair). As a consequence, the equivalence of strongly affine extensions, quasi-Prüfer extensions and INC-pairs is shown. Let G be...

On symmetric semialgebraic sets and orbit spaces

Ludwig Bröcker (1998)

Banach Center Publications

For a symmetric (= invariant under the action of a compact Lie group G) semialgebraic basic set C, described by s polynomial inequalities, we show, that C can also be written by s + 1 G-invariant polynomials. We also describe orbit spaces for the action of G by a number of inequalities only depending on the structure of G.

On the Anderson-Badawi ω R [ X ] ( I [ X ] ) = ω R ( I ) conjecture

Peyman Nasehpour (2016)

Archivum Mathematicum

Let R be a commutative ring with an identity different from zero and n be a positive integer. Anderson and Badawi, in their paper on n -absorbing ideals, define a proper ideal I of a commutative ring R to be an n -absorbing ideal of R , if whenever x 1 ⋯ x n + 1 ∈ I for x 1 , ... , x n + 1 ∈ R , then there are n of the x i ’s whose product is in I and conjecture that ω R [ X ] ( I [ X ] ) = ω R ( I ) for any ideal I of an arbitrary ring R , where ω R ( I ) = min { n : I is an n -absorbing ideal of R } . In the present paper, we use content formula techniques to prove that their conjecture is true, if one of the following conditions...

On the approximate roots of polynomials

Janusz Gwoździewicz, Arkadiusz Płoski (1995)

Annales Polonici Mathematici

We give a simplified approach to the Abhyankar-Moh theory of approximate roots. Our considerations are based on properties of the intersection multiplicity of local curves.

On the arithmetic of arithmetical congruence monoids

M. Banister, J. Chaika, S. T. Chapman, W. Meyerson (2007)

Colloquium Mathematicae

Let ℕ represent the positive integers and ℕ₀ the non-negative integers. If b ∈ ℕ and Γ is a multiplicatively closed subset of ℤ b = ℤ / b ℤ , then the set H Γ = x ∈ ℕ | x + b ℤ ∈ Γ ∪ 1 is a multiplicative submonoid of ℕ known as a congruence monoid. An arithmetical congruence monoid (or ACM) is a congruence monoid where Γ = ā consists of a single element. If H Γ is an ACM, then we represent it with the notation M(a,b) = (a + bℕ₀) ∪ 1, where a, b ∈ ℕ and a² ≡ a (mod b). A classical 1954 result of James and Niven implies that the only ACM...

On the Briançon-Skoda theorem on a singular variety

Mats Andersson, Håkan Samuelsson, Jacob Sznajdman (2010)

Annales de l’institut Fourier

Let Z be a germ of a reduced analytic space of pure dimension. We provide an analytic proof of the uniform Briançon-Skoda theorem for the local ring 𝒪 Z ; a result which was previously proved by Huneke by algebraic methods. For ideals with few generators we also get much sharper results.

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