The main aim of this book is to study the concept of multiplication objects from a categorical point of view, namely, in the setting of monoidal categories which are responsible for the narrow relationship between quantum groups and knot theory. At the same time, the book brings together the literature on multiplication modules and rings, which has been scattered to date. This book organises and exposes them in a categorical framework by using functorial techniques. Multiplication modules and rings are framed inside commutative algebra, which is a basis for number theory and algebraic geometry. These include families of rings very important in ideal arithmetic such as regular von Neumann rings, Dedekind domains, hereditary rings or special primary rings. In the relative case, i.e., multiplication modules and rings with respect to a hereditary torsion theory, the most significant example is that of Krull domains (with respect to the classical torsion theory). As a consequence, we have an adequate setting to consider divisorial properties. As for the graded concept, it is possible to examine deep in the study of arithmetically graded rings such as generalized Rees rings, graded Dedekind domains, twisted group rings, etc. The book points out some different possibilities to deal with the topic, for example, semiring theory, lattice theory, comodule theory, etc.
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