1
Units and Zero Divisors
Units, regular elements, and their equivalence in finite commutative rings
2
Integral Domains
Definitions and inclusions of Euclidean domains, PIDs, and UFDs
3
Chinese Remainder Theorem
Chinese remainder theorem for comaximal ideals
4
Polynomial Rings
Factorization in polynomial rings over commutative rings and Gauss’s lemma
5
Division Rings
Division rings, quaternions, Wedderburn’s little theorem, and Schur’s lemma
6
Central Idempotents and Ring Decomposition
Central idempotents, ring product decomposition, and the Chinese remainder theorem
7
Semisimple Modules
Equivalent characterizations of semisimple modules, isotypic decomposition, and uniqueness of decomposition
8
Artin-Wedderburn Theorem
Structure theorem for semisimple rings and unique decomposition into matrix rings