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The notes in the sections "Basic ring theory" and "Basic module theory" are based on the Math 371 course by Professor Sveta Makarova. I find her videos very clear and easy to follow. The following link will take you to the video playlist.
Professor Sveta Makarova' online course on rings, fields and Galois theory
My notes are self-contained markdown (.md) files. Please view these in a markdown renderer that supports TeX grammar.
These lessons start from the basic concept of a module, and proceeds to reveal the connection between the structure of finitely generated Abelian groups and the the existence of Jordan canonical forms. Familiarity with materials in the section "Basic ring theory" and some very basic group theory is neeeded.
Lesson 1. The concept of a "module"