Strong right D-domains
type: Item
instance of: scholarly article
title: Strong right D-domains
zbMATH Open document ID: 0679.16004
DOI: 10.1007/BF01300343
published in: Monatshefte für Mathematik
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178421
review text: An integral domain R (i.e. an associative ring without zerodivisors having a unit-element 1) is called right D-domain if its lattice of right ideals is distributive and commutative right D-domains are exactly the commutative Prüfer domains. In {\S}2 a sufficient condition for an integral domain R is given such that R is a right D-domain if and only if R is a left D-domain. For example each integral domain which is algebraic over its center satisfies this criterion. Furthermore, a right D-domain is called strong if its lattice of all fractional right ideals \({\mathcal R}\) is distributive. Examples of strong right D-domains are given in {\S}4. Each overring of a strong right D-domain is also a strong right D-domain whereas arbitrary right D-domains may have overrings which are not right D-domains. {\S}3 is mainly concerned with the set \({}^*{\mathcal R}\) of all left invertible fractional right ideals and the mapping \(\lambda\) : \({}^*{\mathcal R}\to {\mathcal L}^*\), \(I\mapsto I_ l^{-1}\) where \(I_ l^{-1}\) denotes the left inverse of I. For example, equivalent conditions are given for \({}^*{\mathcal R}\) to be a sublattice of \({\mathcal R}\) and it is shown that \(\lambda\) is bijective if and only if \(\lambda (I\cap J)=\lambda (I)+\lambda (J)\) holds for all \(I,J\in^*{\mathcal R}\). Finally, {\S}5 deals with (right) D-domains which are algebraic over their centers C. It is proved that C is invariant if and only if C is a commutative Prüfer domain and R the integral closure of C in Q(R).
Mathematics Subject Classification ID: 16U30
Mathematics Subject Classification ID: 16U10
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 13F05
zbMATH DE Number: 4112800
zbMATH Keywords: lattice of right ideals
zbMATH Keywords: right D-domains
zbMATH Keywords: Prüfer domains
zbMATH Keywords: integral domain
zbMATH Keywords: fractional right ideals
zbMATH Keywords: strong right D-domains
zbMATH Keywords: left invertible fractional right ideals
author: Joachim Graeter
MaRDI profile type: Publication
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OpenAlex ID: W2373433399
description: scientific article; zbMATH DE number 4112800