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The present book contains a complete and rigorous treatment of Gröbner bases for modules over commutative polynomial rings with coefficients in Noetherian rings (with some other natural computational conditions), and shows also non-trivial applications of this theory in homological algebra. Algorithmic proofs of some classical theorems of homological algebra using Gröbner bases and matrix constructive methods have been published in many recent papers, but there is not a book that contains both topics. In fact, probably there is not a monograph that simultaneously includes the theory of Gröbner and also presents constructive proofs of three key theorems: Hilbert s Syzygy Theorem, Serre s Theorem, and Quillen-Suslin Theorem. The main purpose of this book is to fill this lack. Some generalizations of these theorems to extended modules and rings from a constructive approach are also included.