Rings, Modules and the Total is an introduction to the theory of rings and modules, concentrating on decomposition properties. The authors introduce the concept of the total and develop its theory, techniques and several intermediate results which allow to prove a number of decomposition theorems.
For those who have never encountered the total before (I had not), we include here a definition. The partially invertible elements of a ring R (with 1) are the divisors of the idempotents of the ring. The total of R, denoted by Tot(R), is the set of al elements of R which are not partially invertible. Let M and W be right R-modules and f an element of Hom(M,W). We say that f is partially invertible if there exits a g in Hom(W,M) such that the composition gf is an idempotent in End(M). The total on M to W, denoted by Tot(M,W), is then defined to be the set of all elements of Hom(M,W) which are not partially invertible.
The concept of the total was first introduced by one of the authors, F. Kasch, in 1982. Since then other authors (W. Schneider, Beidar, Wiegand, Zelmanowitz, Zollner) have published results related to the total, most of which are explained in the book being reviewed.
The book starts from the basic (and more conventional) isomorphism and decomposition theorems for rings and modules. Soon the total is introduced and the basic properties are presented. The next chapter explores which are the "good conditions" for the total (notice that in general the total is not closed under addition, however, under some "good" conditions the total is additively closed). The last two chapters concentrate on LE-decompositions and the total in torsion-free Abelian groups. As an example, we include here the statement of one of the main theorems proved in the book:
Theorem (IV 3.3): Assume that M is a module with an LE-decomposition, and set S = End(M). Then the quotient ring S/Tot(S) is isomorphic to a product of endomorphism rings of vector spaces over division rings.
The book is self-contained, well organized and nicely written, making it a very effective introduction to the subject at hand: the total. Only a basic understanding of ring and module theory is required from the reader (a year of graduate-level abstract algebra would suffice). The book includes proofs for even the elementary results. Perhaps some exercises would have been useful to the reader (none are proposed), hopefully these will be included in future editions of the volume.
Álvaro Lozano-Robledo is H. C. Wang Assistant Professor at Cornell University.