Classification of modules over laterally complete regular algebras

TL;DR

This paper introduces a classification method for modules over laterally complete regular algebras using a passport invariant, which uniquely characterizes module isomorphism classes.

Contribution

It defines a new invariant called passport for modules over laterally complete regular algebras and proves it completely classifies module isomorphism.

Findings

The passport $\Gamma(X)$ uniquely determines module isomorphism.

Modules are classified by partitions of unity and cardinal numbers.

The classification provides a complete invariant for modules over these algebras.

Abstract

Let $\mathcal{A}$ be a laterally complete commutative regular algebra and $X$ be a laterally complete $\mathcal{A}$-module. In this paper we introduce a notion of passport $\Gamma(X)$ for $X$, which consist of uniquely defined partition of unity in the Boolean algebra of idempotents in $\mathcal{A}$ and the set of pairwise different cardinal numbers. It is proved that $\mathcal{A}$-modules $X$ and $Y$ are isomorphic if and only if $\Gamma(X) = \Gamma(Y)$.