Perspectivity in complemented modular lattices and regular rings

TL;DR

This work addresses when isomorphic principal right ideals $aR \cong bR$ in a von Neumann regular ring $R$ imply perspectivity $aR\sim bR$, under the condition that $aR\cap bR$ has finite height in the lattice $L(R)$. It develops a lattice-theoretic reduction via partial isomorphisms and an $\aleph_0$-complete complemented modular lattice framework to transport module isomorphisms into lattice perspectivity, and it derives equational criteria (via terms $t_n,u_n,p_n$) that certify unit-regularity from finite-length data. The paper then connects these lattice results to existence-varieties of regular rings, proving that unit-regularity, direct finiteness, and perspectivity are equivalent across such varieties, and showing equivalent characterizations in terms of artinian generation and explicit identities like $s_{n+1}(x)s_n(x)=s_n(x)$. The findings extend classical results of Ehrlich and Handelman, provide a robust algebraic framework for analyzing unit-regularity in regular rings, and raise open questions for $*$-regular rings, highlighting the role of artinian substructures in governing global regularity properties.

Abstract

Based on an analogue for systems of partial isomorphisms between lower sections in a complemented modular lattice we prove that principal right ideals $aR \cong bR$ in a (von Neumann) regular ring $R$ are perspective if $aR \cap bR$ is of finite height in $L(R)$. This is applied 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 $\mathcal{V}$.

Figures (2)

• Figure 1: Lemma 4
• Figure 2: Lemma 5

Theorems & Definitions (36)

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• Lemma 4
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• Lemma 5
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• Lemma 7
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• Theorem 8
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