Ring Extensions with Finitely Many Non-Artinian Intermediate Rings

Noômen Jarboui,Omar Almallah,Naseam Al-Kuleab,Elena Guardo

doi:10.1155/2020/7416893

Noômen Jarboui, Omar Almallah + Show 2 more

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https://doi.org/10.1155/2020/7416893

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Abstract

The commutative ring extensions with exactly two non-Artinian intermediate rings are characterized. An initial step involves the description of the commutative ring extensions with only one non-Artinian intermediate ring.

Highlights

Journal of Mathematics integral domain, R is a maximal non-Artinian subring of S if and only if R is a rank one valuation domain with quotient field S
We show in eorem 1 that there exists a unique intermediate ring T between R and S such that T is not Artinian if and only if R is a maximal nonArtinian subring of S if and only if R ⊂ S is a closed minimal extension and S is Artinian
Let R ⊂ S be a ring extension. roughout this paper, RS denotes the integral closure of R in S and R′ denotes the integral closure of R

Summary

Introduction

Let R ⊂ S be a ring extension. roughout this paper, RS denotes the integral closure of R in S and R′ denotes the integral closure of R (in its total quotient ring). Roughout this paper, RS denotes the integral closure of R in S and R′ denotes the integral closure of R (in its total quotient ring). We use “⊆” for inclusion and “⊂ ” for strict inclusion. Any undefined notation or terminology is standard, as in [19, 20]

Ring Extensions with Only One Non-Artinian Intermediate Ring

Ring Extensions with Exactly Two NonArtinian Intermediate Rings