Over the last forty years, interest has arisen in the study of modern algebraic systems like Near-rings. Near-rings got several applications in sister fields like automata, Geometry, those are directly related and applicable to present day developing technology. There are more than one thousand near-ringers (Mathematical Researchers working in the area: "Near-ring Theory") from different countries. Due to this popularity near - rings got occupied a vital place in Algebraic systems. A new concept Gamma near-ring was introduced by the Author Bhavanari Satyanarayana. It paved a way for new researchers of mathematical world to create more interest. This book covers a wider and deeper study of several important concepts like Goldie Dimension, Prime Radical, Primary and Tertiary Decompositions in N-groups and modules over Gamma near-rings. This book is certainly attracts the interest of all near-ringers. This book is a more valuable addition not only to the institutional libraries, but also to each personal library of every algebraist of the world.