1
Invertible Elements and Zero Divisors
Units, regular elements, and when regular elements become units in finite commutative rings
2
Integral Domains
Definitions of Euclidean domains, PIDs, and UFDs, and their inclusion relationships
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