Purity, algebraic compactness, direct sum decompositions, and representation type
Abstract
This survey article is devoted to the notions of purity, algebraic and -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 (-)algebraically compact modules are presented, as well as the functorial underpinnings on which they are based. In particular, product-compatible functors, matrix functors and -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.
Cite
@article{arxiv.1407.2360,
title = {Purity, algebraic compactness, direct sum decompositions, and representation type},
author = {Birge Huisgen-Zimmermann},
journal= {arXiv preprint arXiv:1407.2360},
year = {2014}
}