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

This survey article is devoted to the notions of purity, algebraic and $\mathrm{\Sigma }$-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 ($\mathrm{\Sigma }$-)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.