Relative Homological Algebra

Auslander and Bridger originally conceived relative homological algebra, further developed by Enochs, Jenda, and Torrecillas (see [16, 17, 75, 76, 79]). Between 1993 and 1995, Enochs and Jenda introduced the notions of Gorenstein projective and Gorenstein injective modules and, together with Torrecillas, the concept of Gorenstein flat modules. Post-2004, Holm, Bennis, Mahdou, and others expanded this field by introducing various kinds of Gorenstein homological dimensions, allowing classical homological algebra’s theories and methods to be applied to relative homological algebra. However, the proofs of the corresponding propositions are complex and challenging, often requiring additional conditions. This chapter aims to introduce relative homological algebra, particularly focusing on how concepts related to integral domains, as seen in classical ideal theory, are defined and explored using Gorenstein projective modules.