In this course we study the following subjects : Rings, zero-divisors and units, Polynomial Rings, Matrix Rings, integral domain and fields, Ring Homomorphism., Ideals. (right ideals and left ideals ). Factor Rings, Isomorphism Theorems for rings. Maximal ideals, Prime ideals, Radicals of ideals, Primary ideals .The Chinese Remainder Theorem for rings. Euclidean Domains. Principle Ideal Domains. Unique Factorization Domains, Irreducibility Criteria., Module, submodules, Module Homomorphism. Isomorphism Theorems for modules. Prime submodules , maximal submodules, primary submodules . Generation of modules direct sume and free modules. The Chinese Remainder Theorem for module Noetherian R-modules. Finitely generated R-modules.