Almost perfect domains and their modules

The discussion of almost perfect domains and their modules presented in this survey paper is divided in two parts. In the first part the ring-theoretical properties of almost perfect domains are investigated, and different possibilities concerning the behaviour of their archimedean valuation overrings are shown. Connections of almost perfect domains with other well known classes of integral domains are established. The second part illustrates the influence of the property of a domain R of being almost perfect on the category of R-modules, focusing on the subcategories of torsion modules, flat modules, and divisible modules. A final section summarizes the module-theoretical results in the frame of the cotorsion pairs. The two parts are separated by a section containing concrete examples of different kinds of almost perfect domains. A final section on open questions concludes the paper.