AR02 Rings, Integral Domains, and Fields: Definitions and Basic Properties

What is a ring? What is a commutative ring with identity? What is the group of units of a ring and why is it a group? What do 47x and x^47 mean in a ring? What is a field & a Division ring? What are zero divisors and what is an integral domain? Why are fields integral domains and why do cancellation laws hold in integral domains? Why are finite integral domains fields? Subscribe ‪@Shahriari‬ for math videos at the undergraduate level.#abstractalgebra #ringtheory 00:00 Introduction 00:27 Definition of Rings (   • Видео  ) 3:05 Definition of Rings with identity 3:44 Definition of Commutative Rings 4:52 Proof that addition in a ring with identity has to be commutative 8:16 Proof of Elementary properties of rings 19:56 Definition of Group of Units (the set of invertible elements in a ring) 22:16 Proof that the group of units is indeed a group 24:47 Definitions of Multiplying a ring element by an integer and of exponentiation 30:52 Definition of Fields 31:14 Definition of Division Rings 32:26 Definition of Zero Divisors 36:25 Definition of Integral Domains 38:23 Proof that Fields are integral domains 40:10 Proof that Cancellation laws hold in Integral Domains 43:32 Proof that Finite Integral Domains are Fields Next Video:    • AR03 Examples of Rings and Fields   A series of lectures for a full course on undergraduate abstract algebra based on my book: Shahriar Shahriari, Algebra in Action, A Course in Groups, Rings, and Fields, American Mathematical Society, 2017. https://bookstore.ams.org/amstext-27/ An annotated list of Abstract Algebra Videos: https://pomona.box.com/s/06zu90ewikpn... YouTube Playlist:    • Abstract Algebra, an intro via Actions   Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont, CA USA Shahriari is a 2015 winner of the Mathematical Association of America's Haimo Award for Distinguished Teaching of Mathematics, and six time winner of Pomona College's Wig teaching award. Lectures on ring theory in this series:    • AR01 Why Rings? Diophantine Equations & Rings      • AR02 Rings, Integral Domains, and Fields: ...  ​​    • AR03 Examples of Rings and Fields  ​    • AR04 Subrings and their identities      • AR05 Ring Homomorphisms, Kernels, and Ideals      • AR06 What is an Ideal of a Ring? Ideals, H...      • AR07 Generating Ideals. What is a Principa...      • AR08 Zorn's Lemma: Basis for Vector spaces...