This book deals with distributive and semidistributive modules and rings (a module \(M\) is distributive if its lattice of submodules is distributive and \(M\) is semidistributive if it is a direct sum of distributive modules). Besides of an account of the present situation of this problem, the author pursues two additional aims: to provide the reader with an introduction to the homological and structural methods of ring theory and to develop some tools, which are not covered in other monographs.
The book contains 12 chapters, which in their total constitute an essential part of the theory of rings and modules, covering a wide range of themes and questions: 1) radicals, local and semisimple modules; 2) projective and injective modules; 3) Bezout and regular modules; 4) continuous and finite-dimensional modules; 5) rings of quotients; 6) flat modules and semiperfect rings; 7) semihereditary and invariant rings; 8) endomorphism rings; 9) distributive rings with maximum conditions; 10) self-injective and skew-injective rings; 11) semidistributive and serial rings; 12) monoid rings and related topics.
Along with the well known results, a large set of new facts is expounded connecting the previous problems with the condition of distributivity. Thus in some sense the book represents a revision of a part of ring theory from the point of view of distributivity. This new aspect permits to supplement known facts or to give them another interpretation. With such inclination many classes of rings and modules are studied, as well as rings of quotients and rings of endomorphisms. The motif of distributivity is combined in a natural way with the traditional problems.
The relations of distributivity with many other conditions (invariance, hereditary, multiplication, etc.) are studied in detail, as well as with the Bezout and serial rings, with prime ideals etc. Some theorems on the homological classification of rings are also reconsidered, taking into account the influence of the distributive condition and skilfully using some variations of injectivity and projectivity. Of special interest are the results on distributivity of modules and rings with finiteness conditions (in particular, such restrictions lead to curious arithmetics of ideals).
The book contains a considerable part of the most well-known results of ring theory (theorems of H. Bass, A. Goldie, C. Faith, etc.) and also the characterizations of basic types of rings (regular, perfect, quasi-Frobenius etc.). This book, written by a leading researcher in the field, is the first monograph which sums up the distributivity subject in the theory of rings and modules. It includes all necessary definitions and proofs, the presentation is selfcontained and accessible, and so can be used both as a textbook and as an up-to-date reference to the field.