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

Jaballah, Ali

doi:10.1007/s10587-010-0002-x

Cited by 16 publications

(2 citation statements)

References 11 publications

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“…domains is called a normal pair, as defined in [6], if each intermediate ring is integrally closed in B. If A ⊆ B satisfies FIP and A is integrally closed in B, then (A, B) is a normal pair (see [13]).…”

Section: Introductionmentioning

confidence: 99%

From Topologies of a Set to Subrings of Its Power Set

Jaballah

Jarboui

2020

Bull. Aust. Math. Soc.

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Let $X$ be a nonempty set and ${\mathcal{P}}(X)$ the power set of $X$. The aim of this paper is to identify the unital subrings of ${\mathcal{P}}(X)$ and to compute its cardinality when it is finite. It is proved that any topology $\unicode[STIX]{x1D70F}$ on $X$ such that $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$, where $\unicode[STIX]{x1D70F}^{c}=\{U^{c}\mid U\in \unicode[STIX]{x1D70F}\}$, is a unital subring of ${\mathcal{P}}(X)$. It is also shown that $X$ is finite if and only if any unital subring of ${\mathcal{P}}(X)$ is a topology $\unicode[STIX]{x1D70F}$ on $X$ such that $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$ if and only if the set of unital subrings of ${\mathcal{P}}(X)$ is finite. As a consequence, if $X$ is finite with cardinality $n\geq 2$, then the number of unital subrings of ${\mathcal{P}}(X)$ is equal to the $n$th Bell number and the supremum of the lengths of chains of unital subalgebras of ${\mathcal{P}}(X)$ is equal to $n-1$.

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Section: Introductionmentioning

confidence: 99%

From Topologies of a Set to Subrings of Its Power Set

Jaballah

Jarboui

2020

Bull. Aust. Math. Soc.

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“…⊂ R n = S is finite and results from juxtaposing n minimal extensions R i ⊂ R i+1 , 0 ≤ i ≤ n − 1. The FIP property was introduced and studied in [1] and, along with various related properties, has been treated in many other papers [2][3][4][5][8][9][10][11]. In particular, Section 3 of [1] was devoted to the study of ring extension R ⊂ S satisfying FIP when R is a field.…”

Section: Introductionmentioning

confidence: 99%

A note on the FIP property for extensions of commutative rings

Mabrouk

Zeidi

2019

Filomat

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A ring extension R ⊂ S is said to be FIP if it has only finitely many intermediate rings between R and S. The main purpose of this paper is to characterize the FIP property for a ring extension, where R is not (necessarily) an integral domain and S may not be an integral domain. Precisely, we establish a generalization of the classical Primitive Element Theorem for an arbitrary ring extension. Also, various sufficient and necessary conditions are given for a ring extension to have or not to have FIP, where S = R[α] with α a nilpotent element of S.

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Numerical characterizations of some integral domains

Jaballah

2010

Monatsh Math

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

Cited by 16 publications

References 11 publications

From Topologies of a Set to Subrings of Its Power Set

From Topologies of a Set to Subrings of Its Power Set

A note on the FIP property for extensions of commutative rings

Numerical characterizations of some integral domains

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