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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12_2025_894_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$aR \cong bR$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mi>R</mi> <mo>≅</mo> <mi>b</mi> <mi>R</mi> </mrow> </math></EquationSource> </InlineEquation> in a (von Neumann) regular ring R are perspective if the series becomes stationary. In particular, this applies if <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12_2025_894_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$aR \cap bR$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mi>R</mi> <mo>∩</mo> <mi>b</mi> <mi>R</mi> </mrow> </math></EquationSource> </InlineEquation> is of finite height in L(R). This is used to derive, for existence-varieties <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12_2025_894_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {V}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">V</mi> </math></EquationSource> </InlineEquation> of regular rings, equivalence of unit-regularity and direct finiteness, both conceived as a property shared by all members of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12_2025_894_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {V}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">V</mi> </math></EquationSource> </InlineEquation>.

Perspectivity in complemented modular lattices and regular rings

期刊论文
发表日期：

2025年6月19日

查看全文

Christian Herrmann

摘要

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 $aR \cong bR$ $a R ≅ b R$ in a (von Neumann) regular ring R are perspective if the series becomes stationary. In particular, this applies if $aR \cap bR$ $a R \cap b R$ is of finite height in L(R). This is used to derive, for existence-varieties $\mathcal {V}$ $V$ of regular rings, equivalence of unit-regularity and direct finiteness, both conceived as a property shared by all members of $\mathcal {V}$ $V$ .