Abstract Algebra - The Basics of Rings

Learn the fundamental concepts of ring theory in abstract algebra through this comprehensive video series covering essential definitions, properties, and theorems. Explore what constitutes a ring and examine various types including integral domains, while working through basic results and exercises to solidify understanding. Master the concepts of units and zero divisors, then delve into ideals and ring isomorphisms with concrete examples. Discover the characteristic of a ring and understand ring homomorphisms, including their motivation and properties. Study principal ideals and their significance in ring theory, then progress to the First and Second Isomorphism Theorems for rings. Examine maximal and prime ideals, their definitions and applications, before concluding with polynomial rings and additional ring theory examples to reinforce the material covered throughout the series.

Syllabus

Abstract Algebra | What is a ring?
Abstract Algebra | Types of rings.
Abstract Algebra | Some basic results regarding integral domains.
Abstract Algebra | Some basic exercises involving rings.
Abstract Algebra | Units and zero divisors of a ring.
Abstract Algebra | More examples involving rings: ideals and isomorphisms.
Abstract Algebra | The characteristic of a ring.
Abstract Algebra | Ring homomorphisms
Abstract Algebra | The motivation for the definition of an ideal.
Abstract Algebra | Principal Ideals of a Ring
Abstract Algebra | Properties and examples of ring homomorphisms.
Abstract Algebra | First Isomorphism Theorem for Rings
Abstract Algebra | Maximal and prime ideals.
Abstract Algebra | The Second Isomorphism Theorem for Rings
Abstract Algebra | More ring theory examples.
Abstract Algebra | Polynomial Rings