this is an account of how a certain fundamental algebraic concept can be introduced, developed, and applied to solve some concrete algebraic problems. The book is divided into three parts. The first is concerned with defining concepts and terminology, assembling elementary facts, and developing the theory of factorization in a principal ideal domain. The second part deals with the main decomposition theorems which describe the structure of finitely generated modules over a principal ideal domain. The third part contains the applications of these theorems. This book may be of interest to undergraduates taking courses in algebra.