Flat modules and rings finitely generated as modules over their center
type: Item
instance of: scholarly article
title: Flat modules and rings finitely generated as modules over their center
zbMATH Open document ID: 0899.16001
DOI: 10.1007/BF02305182
published in: Mathematical Notes
publication date: 1998-11-15T00:00:00Z
review text: The distributivity of the ring \(A\) is investigated in the case when \(A\) satisfies the condition: (*) \(A\) is a finitely generated module over a central unitary subring \(R\). This article contains many facts on such rings, in particular the relations with rings of quotients and heredity of classes of flat modules are revealed. Among the main results the following two theorems are decisive. I. For a ring \(A\) with (*) the following conditions are equivalent: 1) \(A\) is a right distributive semiprime ring; 2) \(A\) is a left distributive semiprime ring; 3) for any ideal \(M\in\max(R)\) the ring of quotients \(A_M\) is the finite direct product of semihereditary uniform Bezout domains whose quotient rings by their Jacobson radicals are finite direct products of skew fields. II. If \(A\) is a left distributive ring with (*) then: 1) \(A\) is a distributive reduced ring over which all submodules of flat right modules and flat left modules are flat; 2) any prime quotient ring of the ring \(A\) is a semihereditary order in a skew field.
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16H05
zbMATH DE Number: 1158890
zbMATH Keywords: distributivity of rings
zbMATH Keywords: finitely generated modules
zbMATH Keywords: rings of quotients
zbMATH Keywords: flat modules
zbMATH Keywords: right distributive semiprime rings
zbMATH Keywords: finite direct products of skew fields
zbMATH Keywords: prime quotient rings
zbMATH Keywords: semihereditary orders
author: A. A. Tuganbaev
reviewed by: A. I. Kashu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1158890