Relative FBN rings and the second layer condition
type: Item
instance of: scholarly article
title: Relative FBN rings and the second layer condition
zbMATH Open document ID: 0946.16015
DOI: 10.1016/S0022-4049(97)00192-8
author: Paul Kim
author: Günter Krause
published in: Journal of Pure and Applied Algebra
publication date: 2000-10-22T00:00:00Z
review text: A prime ideal \(P\) of a right Noetherian ring \(R\) satisfies the right second layer condition if the ``second layer'' of the injective hull \(E=E_R(R/P)\) of the right \(R\)-module \(R/P\) is ``tame'' -- that is, every prime submodule of \(E/l\text{-ann}_E(P)\) is torsion-free. This condition was introduced 20 years ago by \textit{A. V. Jategaonkar} [Localization in Noetherian Rings. Lond. Math. Soc. Lect. Note Ser. 98, Cambridge University Press (1986; Zbl 0589.16014)] and the reviewer [\textit{K. A. Brown}, J. Algebra 69, 247--260 (1981; Zbl 0471.16012)], and has been much studied since then. The ring \(R\) satisfies the right second layer condition if all its prime ideals do. In this paper the authors compare and discuss several variants of this definition which have appeared in the literature. In their main result (Theorem 6.2) they provide various equivalent formulations of the condition, in particular giving several such formulations not involving any explicit mention of injective hulls. In doing so they define a new dimension for a module \(M\), the tame dimension, defined as the deviation of the set of submodules \(T\) of \(M\) with \(M/T\) tame, which may have independent applications.
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 16P60
zbMATH DE Number: 1308263
zbMATH Keywords: fully bounded Noetherian rings
zbMATH Keywords: right Noetherian rings
zbMATH Keywords: second layer condition
zbMATH Keywords: injective hulls
zbMATH Keywords: prime ideals
zbMATH Keywords: tame dimension
reviewed by: Kenneth A. Brown
MaRDI profile type: Publication
cites work: Localization at collections of minimal primes
cites work: Les t-anneaux, la condition (H) de gabriel et ses consequences
cites work: Q3994888
cites work: Relative krull dimension and prime ideals in right noetherian rings
cites work: Solvable lie algebras polycyclic-by-finite groups and bimodule krull dimension
cites work: Q3717234
cites work: Module extensions and the second layer condition
cites work: Q4891433
cites work: Q5682125
cites work: Q3786586
cites work: Localization and right krull links
full work available at URL: s0022-4049(97)00192-8
OpenAlex ID: W1987004147
description: scientific article; zbMATH DE number 1308263