This course is primarily designed for students majored in pure and applied mathematics, and is divided into Parts I and II, lasting ideally for two consecutive semesters. It aims at leading students into systematic and thorough studies of the fundamentals of modern algebra, thus getting them to lay a solid foundation for subsequent, more advanced courses in math major. The contents of the course and the standards of assessment will normally surpass the other courses in the same series, the objective being to foster students with best background in algebra.