On strongly affine extensions of commutative rings

Nabil Zeidi

Czechoslovak Mathematical Journal (2020)

  • Volume: 70, Issue: 1, page 251-260
  • ISSN: 0011-4642

Abstract

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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 a subgroup of the automorphism group of S such that R is invariant under action by G . If R ⊆ S is strongly affine, then R G ⊆ S G is strongly affine under some conditions.

How to cite

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Zeidi, Nabil. "On strongly affine extensions of commutative rings." Czechoslovak Mathematical Journal 70.1 (2020): 251-260. <http://eudml.org/doc/296978>.

@article{Zeidi2020,
abstract = {A ring extension $R\subseteq S$ 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 $R\subseteq S$ is integrally closed and strongly affine if and only if $R\subseteq S$ 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 a subgroup of the automorphism group of $S$ such that $R$ is invariant under action by $G$. If $R\subseteq S$ is strongly affine, then $R^G\subseteq S^G$ is strongly affine under some conditions.},
author = {Zeidi, Nabil},
journal = {Czechoslovak Mathematical Journal},
keywords = {strongly affine; Prüfer extension; finitely many intermediate algebras property extension; finite chain propery extension; normal pair; integrally closed pair; ring of invariants},
language = {eng},
number = {1},
pages = {251-260},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On strongly affine extensions of commutative rings},
url = {http://eudml.org/doc/296978},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Zeidi, Nabil
TI - On strongly affine extensions of commutative rings
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 1
SP - 251
EP - 260
AB - A ring extension $R\subseteq S$ 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 $R\subseteq S$ is integrally closed and strongly affine if and only if $R\subseteq S$ 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 a subgroup of the automorphism group of $S$ such that $R$ is invariant under action by $G$. If $R\subseteq S$ is strongly affine, then $R^G\subseteq S^G$ is strongly affine under some conditions.
LA - eng
KW - strongly affine; Prüfer extension; finitely many intermediate algebras property extension; finite chain propery extension; normal pair; integrally closed pair; ring of invariants
UR - http://eudml.org/doc/296978
ER -

References

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