Lessons on the Grothendieck category \(\sigma[M]\). (Q2881308)

In the study of module theory it turns out that often one gets a better insight and degree of generality when dealing with a special full subcategory of the entire category of modules \(R\)-Mod, namely the full subcategory \(\sigma[M]\) subgenerated by a given module \(M\). This can also be viewed as the smallest full subcategory of \(R\)-Mod which contains \(M\) and is a Grothendieck category. The most exhaustive presentation of the category \(\sigma[M]\) is given by \textit{Robert Wisbauer}'s well-known monograph [Foundations of module and ring theory. A handbook for study and research. Algebra, Logic and Applications 3. Philadelphia: Gordon and Breach Science Publishers (1991; Zbl 0746.16001)].NEWLINENEWLINENEWLINEThe book under review has grown from a graduate course given by the author at the Department of Pure and Applied Mathematics of the University of Padua, Italy, in 1994. It is organized as a series of 11 lessons, each of them including a short collection of exercises completing and/or illustrating by concrete examples the topics presented. The material is mainly based on Wisbauer's above cited monograph, sometimes expanding or detailing properties, but it also discusses topics such as the Gabriel-Popescu Theorem or the Teply-Miller Theorem. The author has succeeded to make a relevant selection of the highlights of the theory of the category \(\sigma[M]\) together with some of the most important of its applications in module theory. The book is nicely and accurately written, with care for the intended readers, either graduate students or interested researchers.