This course is designed to give the first year graduate students in mathematics a study of more advanced topics on groups, rings, modules, polynomials, Galois theory, commutative algebra, and multilinear algebra as a background for studying number theory and algebraic geometry. Topics include Sylow theory, the fundamental theorem of finitely generated abelian groups, free groups, commutative rings, localization, principal and factorial rings, Notherian rings, free modules, group rings, direct and inverse limits, and some advanced theory of ideals from the theory of rings and modules.