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The present book is the culmination of honest and sincere efforts on the part of the author to meet the requirements of students who opt for mathematics at the Post graduation level. It is also meant for those who are appearing in various competitive examinations. The entire subject matter has been arranged in a simple, systematic, graded, lucid and exhaustive manner to meet the various needs of all types of learners. In order to give the students a taste of UGC-CSIR NET examination, some MCQ’s from the previous years papers and other similar questions have been added at the end of the book.Contents : Factorization Theory in Integral Domains, Divisibility, Unique Factorization Domain (UFD), Principal Ideal Domain (PID), Euclidean Domain (ED) and their relationships, Noetherian and Artinian Rings, Examples and Counter Examples, Artinian Rings without zero divisors, Nil ideals in Artinian Rings, Hilbert Basis Theorem.Modules, Differences between Modules and Vector Spaces, Module Homomorphisms, Quotient Modules, Completely Reducible or Semi-simple Modules, Free Modules, Representation and Rank of Linear Mappings, Smith Normal Form over a PID, Finitely Generated Modules over a PID, Rational Canonical Form, Applications to finitely generated abelian groups.