Visual Algebra, Lecture 8.4: Ring homomorphisms скачать в хорошем качестве

Visual Algebra, Lecture 8.4: Ring homomorphisms

We start by formalizing the notion of a quotient ring. Because a ring is an additive abelian group, the set of cosets of a subring is an quotient group. It is also a ring if multiplication of cosets is well-defined, and this happens precisely when the subring is a two-sided ideal. We’ll prove this, and motivate it with a familiar example. Then, we'll formally define ring homomorphisms, which are just structure-preserving maps between rings. Since rings are additive abelian groups, a ring homomorphism is a group homomorphism that also preserves multiplication. We’ll see some examples of this, and prove some basic properties, such as how the kernel is always a 2-sided ideal. Course & book webpage (with complete lecture note slides, HW, exams, etc.): https://www.math.clemson.edu/~macaule... ------------------------------------------------------------------------------------------------------------------------------------------------------ 0:00 Introduction 1:06 Group theory vs. ring theory 4:44 The quotient ring ℤ₃²/(10) 7:02 Failure of the quotient of ℤ₃² by non-ideals 9:03 Quotient rings 16:37 Ring homomorphism: definition and examples 22:04 Isomorphism theorem prerequisites 30:00 Embeddings and fields 36:16 Preview to the ring isomorphism theorems