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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 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...

Polynômes de Barsky

Youssef Haouat, Fulvio Grazzini (1979)

Annales scientifiques de l'Université de Clermont. Mathématiques

Ring extensions with some finiteness conditions on the set of intermediate rings

Ali Jaballah (2010)

Czechoslovak Mathematical Journal

A ring extension R ⊆ S is said to be FO if it has only finitely many intermediate rings. R ⊆ S is said to be FC if each chain of distinct intermediate rings in this extension is finite. We establish several necessary and sufficient conditions for the ring extension R ⊆ S to be FO or FC together with several other finiteness conditions on the set of intermediate rings. As a corollary we show that each integrally closed ring extension with finite length chains of intermediate rings is necessarily a normal pair...

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