{"title":"Perspectivity in complemented modular lattices and regular rings","authors":"Christian Herrmann","doi":"10.1007/s00012-025-00894-8","DOIUrl":null,"url":null,"abstract":"<div><p>Based on an analogue for systems of partial isomorphisms between lower sections in a complemented modular lattice we construct a series of terms (including inner inverse as basic operation and providing descending chains) such that principal right ideals <span>\\(aR \\cong bR\\)</span> in a (von Neumann) regular ring <i>R</i> are perspective if the series becomes stationary. In particular, this applies if <span>\\(aR \\cap bR\\)</span> is of finite height in <i>L</i>(<i>R</i>). This is used to derive, for existence-varieties <span>\\(\\mathcal {V}\\)</span> of regular rings, equivalence of unit-regularity and direct finiteness, both conceived as a property shared by all members of <span>\\(\\mathcal {V}\\)</span>.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"86 3","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra Universalis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00012-025-00894-8","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Based on an analogue for systems of partial isomorphisms between lower sections in a complemented modular lattice we construct a series of terms (including inner inverse as basic operation and providing descending chains) such that principal right ideals \(aR \cong bR\) in a (von Neumann) regular ring R are perspective if the series becomes stationary. In particular, this applies if \(aR \cap bR\) is of finite height in L(R). This is used to derive, for existence-varieties \(\mathcal {V}\) of regular rings, equivalence of unit-regularity and direct finiteness, both conceived as a property shared by all members of \(\mathcal {V}\).
期刊介绍:
Algebra Universalis publishes papers in universal algebra, lattice theory, and related fields. In a pragmatic way, one could define the areas of interest of the journal as the union of the areas of interest of the members of the Editorial Board. In addition to research papers, we are also interested in publishing high quality survey articles.