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.
CITATION STYLE
Huisgen-Zimmermann, B. (2000). Purity, Algebraic Compactness, Direct Sum Decompositions, and Representation Type. In Infinite Length Modules (pp. 331–367). Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-8426-6_18
Mendeley helps you to discover research relevant for your work.