Noncommutative Prüfer rings satisfying a polynomial identity
type: Item
instance of: scholarly article
title: Noncommutative Prüfer rings satisfying a polynomial identity
zbMATH Open document ID: 0806.16027
author: Patrick Morandi
published in: Journal of Algebra
publication date: 1995-02-13T00:00:00Z
review text: A prime Goldie ring \(R\) is a right Prüfer ring if for every finitely generated right \(R\)-ideal \(I\), \(I^{-1}I = R\) and \(II^{-1} = O_ l(I)\). \(R\) is a left Prüfer ring if for every finitely generated left \(R\)-ideal \(J\) of \(R\), \(JJ^{-1} = R\) and \(J^{-1}J = O_ r(R)\). \textit{J. H. Alajbegovic} and \textit{N. I. Dubrovin} [J. Algebra 135, 165-176 (1990; Zbl 0718.16023)] have shown that a ring is right Prüfer if and only if it is left Prüfer. The paper investigates the relation of Prüfer rings to other types of rings as e.g. valuation rings (in the sense of Dubrovin) and Bezout rings (rings in which every finitely generated right ideal is principal). The main theorems are: (Theorem 1.9) Let \(R\) be Prüfer and \(B_ 1,\dots,B_ n\) be valuation (Dubrovin) overrings of \(R\) then the \(B_ i\) satisfy the intersection property, i.e. \(Q \mapsto Q \cap R\) gives a one-to-one correspondence between the prime ideals of the various \(B_ i\) and the prime ideals of \(R\). (Theorem 2.5) Let \(R\) be a prime PI-ring with center \(A\). Suppose \(A\) is Prüfer. Then \(R\) is a Prüfer ring in which each maximal ideal is localizable if and only if \(R_ m\) is Bezout for all maximal ideals \(m\) of \(A\).
reviewed by: Jan Van Geel
Mathematics Subject Classification ID: 16R20
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16U20
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 554245
zbMATH Keywords: prime Goldie ring
zbMATH Keywords: right Prüfer ring
zbMATH Keywords: left Prüfer ring
zbMATH Keywords: valuation rings
zbMATH Keywords: Bezout rings
zbMATH Keywords: finitely generated right ideal
zbMATH Keywords: intersection property
zbMATH Keywords: prime ideals
zbMATH Keywords: prime PI-ring
zbMATH Keywords: center
zbMATH Keywords: maximal ideal
MaRDI profile type: Publication
full work available at URL: jabr.1993.1222
OpenAlex ID: W2075562461
DOI: 10.1006/JABR.1993.1222
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description: scientific article; zbMATH DE number 554245