Visual Algebra, Lecture 8.5: The ring isomorphism theorems скачать в хорошем качестве

Visual Algebra, Lecture 8.5: The ring isomorphism theorems

The four isomorphism theorems for rings are analogous to the group isomorphism theorems. Loosely speaking, they say that every homomorphic image is a quotient, and then they characterize the structure of a quotient. One of the big themes in group theory is that taking a quotient amounts to chopping off the subgroup lattice, and this preserves the key structural features, such as index, conjugacy classes, normal subgroups, and more. This theme is also true in ring theory, but the key structural features are a little different. We’ll see how taking a quotient preserves the ring substructures—ideals, subrings, and subgroups, which we denote by colors in the subring lattice. Course & book webpage (with complete lecture note slides, HW, exams, etc.): https://www.math.clemson.edu/~macaule... ------------------------------------------------------------------------------------------------------------------------------------------------------ 0:00 Introduction 1:10 Summary of the isomorphism theorems 3:43 The fundamental homomorphism theorem (FHT) 7:43 A commutative diagram illustrating the FHT 9:12 ℤ₂³→ℤ₂²: The FHT and additive Cayley tables 10:27 ℤ₂³→ℤ₂²: The FHT and multiplicative Cayley tables 10:58 The correspondence theorem 12:50 The shoebox analogy of the correspondence theorem 15:02 The correspondence theorem, formally 17:18 A subring lattice interpretation of the correspondence theorem 23:15 The fraction theorem 25:07 A subring lattice interpretation of the fraction theorem 23:03 The shoebox analogy of the fraction theorem for ℤ₈×ℤ₂ 28:48 The shoebox analogy of the fraction theorem for ℤ₆×ℤ₄ 30:15 The diamond theorem: statement and proof 34:44 The diamond theorem and the subring lattice of ℤ₆×ℤ₂ 35:24 The diamond theorem and the subring lattice of ℤ₆×ℤ₄ 36:10 Pizza diagrams and the diamond theorem 37:50 Every homomorphism factors as a quotient and an embedding