Almost free modules, perfect decomposition and Enochs's conjecture

TL;DR

This paper explores advanced module theory concepts, constructing specific modules under set-theoretic assumptions, characterizing perfect decompositions, and demonstrating the consistency of Enoch's conjecture with ZFC.

Contribution

It extends previous results by constructing modules with particular freeness and separability properties for singular cardinals and characterizes when modules have perfect decompositions.

Findings

Constructed non-trivial modules with freeness and separability properties.

Characterized conditions for modules to have perfect decompositions.

Showed Enoch's conjecture is consistent with ZFC.

Abstract

Given a module $X$ and a regular cardinal $\kappa$ we study various notions of $(\kappa,\mathrm{Add}(X))$-freeness and $(\kappa,\mathrm{Add}(X))$-separability. Bearing on appropriate set-theoretic assumptions, we construct a non-trivial $\kappa^+$-generated, $(\kappa^+,\mathrm{Add}(X))$-free and $(\kappa^+,\mathrm{Add}(X))$-separable module. Our construction allows $\kappa$ to be singular thus extending \cite[Theorem~4.7]{CortesGuilTorrecillas}. Bearing on similar set-theoretic assumptions, we characterize when every module $X$ has a perfect decomposition. As a subproduct we show that Enoch's conjecture for classes $\mathrm{Add}(X)$ is consistent with ZFC -- a fact first proved by \v{S}aroch \cite{Saroch}.