The Jacobson radical of the Laurent series ring.
type: Item
instance of: scholarly article
title: The Jacobson radical of the Laurent series ring.
zbMATH Open document ID: 1161.16013
published in: Journal of Mathematical Sciences (New York)
publication date: 2008-10-28T00:00:00Z
review text: All rings in this paper are associative with nonzero identity element. A ring \(A\) is called a ring of bounded index, if there exists a positive integer \(n\) such that \(a^n=0\) for every nilpotent element \(a\in A\). A module is said to be finite-dimensional if it does not contain an infinite direct sum of nonzero submodules. A right finite-dimensional ring with the maximum condition on right annihilators is called a right Goldie ring. For a ring \(A\), the intersection of all prime ideals of \(A\) is called the prime radical of \(A\). A ring \(A\) is said to be semiprime if \(A\) does not have nonzero nilpotent ideals. The intersection of all maximal submodules of the module \(M\) is denoted by \(J(M)\) and is called the Jacobson radical of the module \(M\). A module \(M\) is said to be semiprimitive if \(J(M)=0\). For a ring \(A\) and an automorphism \(\varphi\) of \(A\), the author considers the skew Laurent series ring \(A(\!(x,\varphi)\!)\) with coefficient ring \(A\) formed by all series \(f=\sum_{i=k}^{+\infty}f_ix^i\), where \(x\) is a variable, \(k\) is an integer (possibly negative), and all the coefficients \(f_i\) are contained in \(A\). In the ring \(A(\!(x,\varphi)\!)\), addition is naturally defined and multiplication is defined by the formula \(xa=\varphi(a)x\) for all \(a\in A\). So, if \(\varphi=1_A\), we obtain the ordinary Laurent series ring \(A(\!(x)\!)\). For every subset \(N\) of \(A\), \(N(\!(x,\varphi)\!)\) denotes the subset of the ring \(A(\!(x,\varphi)\!)\) consisting of series \(f=\sum_{i=k}^{+\infty}f_ix^i\) whose coefficients \(f_i\) are contained in \(N\). The author shows that if \(A\) is a ring such that the prime radical \(N\) of \(A\) is nilpotent and the factor ring \(A/N\) is a right Goldie ring (for example, this is the case if \(A\) is a right Noetherian ring), then for every automorphism \(\varphi\) of the ring \(A\), the Jacobson radical of the skew Laurent series ring \(A(\!(x,\varphi)\!)\) is nilpotent and it coincides…
reviewed by: Halina France-Jackson
Mathematics Subject Classification ID: 16N20
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16N40
Mathematics Subject Classification ID: 16D25
zbMATH DE Number: 5358152
zbMATH Keywords: skew Laurent series rings
zbMATH Keywords: prime radical
zbMATH Keywords: Jacobson radical
zbMATH Keywords: right Goldie rings
zbMATH Keywords: rings of bounded index
zbMATH Keywords: semiprimitive rings
zbMATH Keywords: prime rings
zbMATH Keywords: semiprime rings
zbMATH Keywords: rings with right Krull dimension
author: A. A. Tuganbaev
MaRDI profile type: Publication
full work available at URL: s10958-008-0057-5
OpenAlex ID: W2053248370
cites work: Q4744417
cites work: Q5682108
cites work: Krull dimension
cites work: Q3128051
cites work: Biregular rings of Laurent series
cites work: Q4761814
cites work: Laurent series rings and pseudo-differential operator rings.
DOI: 10.1007/S10958-008-0057-5
description: scientific article; zbMATH DE number 5358152