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Synopsis Our aim in this book is basically a very simple one: to show how the familiar methods and ideas used in the analysis of the system of "whole numbers" have been and are being extended and generalized to enable more diverse mathematical systems to be analyzed. In particular the concept of factorization or decomposition into "simpler" pieces underlies essentially all of our efforts. However, before we can proceed with this programme of study we must try to establish a common starting point and path of approach. Even before doing this we must agree upon some basic assumptions, terms, and notations, and these are set down in this book. Since the student will have encountered most of this material previously in one form or another he should strive to maintain contact between the various formulations by constructing examples, doing problems, asking questions of his instructors and fellow students, and reading other accounts of the material. Each topic included in the book is important for degree students of Indian universities.