This monograph covers two main topics: approximations of modules and realization of R-algebras as endomorphism algebras of R-modules over a commutative ring R. The first introductory chapter surveys notions such as S-completions, pure-injective modules, locally projective modules and slender modules, notions repeatedly used in the rest of the book. The next seven chapters deal with the existence of C-approximations, that is C-(pre)covers and, dually, C-preenvelopes, for a class C of modules. This problem is approached by employing cotorsion theories, a method that is justified by results on complete and perfect cotorsion pairs. In particular, the long-standing Flat Cover Conjecture, which predicts the existence of flat covers in any module category, is solved positively.
Beyond these applications many nice results are obtained: for example, the equality of certain cotorsion pairs characterize Dedekind, Prüfer domains and almost perfect domains. Three chapters are devoted to tilting and cotilting cotorsion theories. First the tilting case is settled. The authors characterize cotorsion pairs induced by n-tilting modules and prove that a class of modules is tilting if and only if it is of finite type. Moreover, they characterize cotorsion pairs induced by a 1-tilting module and study these pairs over Artin algebras, Dedekind, Prüfer and valuation domains. The theory of tilting cotorsion pairs is applied to describe Matlis localizations and, in chapter 7, to attack finistic dimension conjectures. The dual cotilting case is studied in chapter 8. It is proved that cotilting modules are pure injective and, as in the tilting case, cotilting torsion theories are characterized.
The second part of the book begins with an introduction of some set theory prediction principles: the Diamond Principles, requiring additional set theory axioms, and the Black Box Principles, which are proved in ZFC. In the next chapter it is proved that completeness of certain cotorsion pairs is independent on ZFC+GCH. In chapter 11, the lattice of cotorsion pairs of abelian groups is investigated. Next, the Black Box Principles are applied to realize R-algebras as endomorphism algebras of R-modules satisfying various cardinality and structural properties. Chapter 13 is devoted to the construction of E(R)-algebras, i.e. R-algebras A that are naturally isomorphic to the endomorphism rings of A as left R-modules. The next chapter settles the problem of realizing R-algebras in case the ring R is not an S-ring for any suitable multiplicative set S. In this case different techniques and set theory principles to the Diamond and the Black Box Principles are needed. The last chapter deals with the realization of some particular algebras, namely Leavitt type algebras, as endomorphism algebras of modules. The complexity of these algebras implies the existence of various pathological decompositions of the modules.
The monograph presents recent achievements in two rapidly developing directions in the theory of modules over associative rings. The reader is introduced to a comprehensive theory based on many nice results and ideas and they are encouraged to participate in its future development. I strongly recommend the monograph to anyone who is interested in the modern theory of modules.