Purity, algebraic compactness, direct sum decompositions, and representation type

TL;DR

This survey explores the concepts of purity, algebraic compactness, and their relation to module decompositions and representation type, highlighting foundational theories and recent developments in module category structure.

Contribution

It provides a comprehensive overview of the notions of purity and algebraic compactness, including characterisations, functorial frameworks, and their implications for ring representation theory.

Findings

Characterisations of algebraically compact modules

Connections between pure global dimension and representation type

Discussion of functorial tools like p-functors and matrix functors

Abstract

This survey article is devoted to the notions of purity, algebraic and $\Sigma$-algebraic compactness, direct sum decompositions, and representation type in the category of modules over a ring. It begins with basic definitions, a brief history, and a discussion of global decomposition problems going back to work of K\"othe and Cohen-Kaplansky; these are strongly tied to algebraic compactness. Characterisations of ($\Sigma$-)algebraically compact modules are presented, as well as the functorial underpinnings on which they are based. In particular, product-compatible functors, matrix functors and $p$-functors are discussed. The latter part of the article is devoted to rings of vanishing left pure global dimension, product completeness, endofiniteness, the pure semisimplicity problem and its connection with a strong Artin problem on division ring extensions.