How to Understand Abstract Algebra

Abstract algebra is a term used to describe a mathematical system consisting of a set of elements and binary operations with rules governing their interaction. This is distinct from elementary algebra which teaches the rules for manipulating arithmetic expressions. Algebra is widely used in both theoretical and applied mathematics and will form a large part of a mathematician's education.

Instructions

Difficulty: Challenging

1. Take at least two semesters of algebra if you plan to major in a scientific subject. Three or even four semesters would be better given the wide use of algebra, but this is usually unrealistic. A study of algebra will include modules, rings, groups and fields. However, these concepts are not discrete and must be taught together.
2. Study the use of commutative diagrams throughout the course. They will be encountered early because abstract algebra uses diagram chasing to prove many theorems. This technique consists of finding equivalent map compositions and exploiting their homomorphisms.
3. Begin with simple modules over rings. Include vector spaces with plenty of examples. This should include the concept of finite dimensionality and lead naturally to the Jordan-Holder Theorem.
4. Advance to modules over commutative rings. This stage should cover localization, associated primes and especially polynomial rings. Unique factorization domains also should be included. The Wedderburn Theorem must be included here because it is one of the most fundamental theorems of modern algebra.
5. Complete a course in algebra fundamentals with a study of modules over Dedekind domains and finitely generated modules. This section should also introduce field theory.

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