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Rings, Modules and Linear Algebra
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.
- GenresMathematics
256 pages, Paperback
First published September 1, 1970
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April 18, 2023
I used it as reference book in parallel with an undergraduate course I took in Rings and Modules. The book is divided into three parts: Part I: Rings and Modules- Part II: Direct Decomposition of a Finitely Generated Module Over a Principal Ideal Domain- Part III: Applications to Groups and Matrices. It was prepared initially as a course of lectures for undergraduate students of mathematics at Warwick. The writing is very clear and comprehensive but not lengthy and the exercises are fruitful.
One of the most satisfying results is seeing the application of the direct decomposition of finitely generated modules over a principal ideal domain (achieved in part two of the book) applied to vector spaces (if V is a vector space over a field K we know that the ring of polynomials K[x] is a PID and we smartly view V as a K[x]-module when dealing with an endomorphism T on V as follows: if p(x)=a_0+a_1 x+...+a_n x^n is a polynomial in K[x] we define the action of p(x) on a vector v in V as p(x)v:=p(T)(v)) and hence to matrices, as one could have already seen in a course on Linear Algebra, and to finitely generated abelian groups(any abelian group is a Z-module and Z is a PID), as one could have already seen in a course on Group Theory.
One of the most satisfying results is seeing the application of the direct decomposition of finitely generated modules over a principal ideal domain (achieved in part two of the book) applied to vector spaces (if V is a vector space over a field K we know that the ring of polynomials K[x] is a PID and we smartly view V as a K[x]-module when dealing with an endomorphism T on V as follows: if p(x)=a_0+a_1 x+...+a_n x^n is a polynomial in K[x] we define the action of p(x) on a vector v in V as p(x)v:=p(T)(v)) and hence to matrices, as one could have already seen in a course on Linear Algebra, and to finitely generated abelian groups(any abelian group is a Z-module and Z is a PID), as one could have already seen in a course on Group Theory.
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