Morita Transport, Marked Projective Loci, and Dedekind Classification for C4-Type Conditionss

Abstract

Objectwise Morita invariance of $\Cfour$ is known. We prove the corresponding statements for $\Cfourstar$, semi-weak-CS, and strongly $\Cfourstar$, including the chain joins in the definition of semi-weak-CS. For a module property $\mathcal{Q}$ preserved and reflected by equivalences, we consider the subset $\Vmon_{\mathcal Q}(R)$ of the finitely generated projective monoid $\Vmon(R)$. This marked pair is transported by Morita equivalence. The associated regular-module ring property is Morita invariant if and only if membership in $\Vmon_{\mathcal Q}(R)$ is constant on the order units. The matrix profile satisfies ${\mu }_{\mathcal{Q}}(M_m(R))=\{n\ge 1:mn\in {\mu }_{\mathcal{Q}}(R)\}$. For a nondivision right Ore domain, the profiles of $\Cfour$, $\Cfourstar$, and strongly $\Cfourstar$ are $\{1\}$, whereas the semi-weak-CS profile is ${\mathbb{N}}_{\ge 1}$. Over a nonfield Dedekind domain $D$, the first three marked loci consist of the projectives of rank at most one, while every finitely generated projective is semi-weak-CS. For a finitely generated $D$-module $M$, we prove that every $\Cfour$ defect at rank at most one can be transferred between $M$ and $\tau (M)$, and we construct a defect when the rank is at least two. Hence $M$ is $\Cfour$ if and only if ${\mathrm{rank}}_{D}M\le 1$ and $\tau (M)$ is $\Cfour$; the analogous criterion holds for $\Cfourstar$. Combined with the finite-length torsion criterion, these reductions give invariant-factor tests for the four conditions and criteria for their behaviour under direct sums.