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Description: A ring is a commutative group under addition that has a second operation: multiplication. These generalize a wide variety of mathematical objects like the integers, polynomials, matrices, modular arithmetic, and more. In this video we will take an in depth look at the definition of a ring. Our Abstract Algebra playlist is here: bit.ly/AbstractAlgebraSocratica Be sure to subscribe so you don't miss new lessons from Socratica: bit.ly/1ixuu9W ♦♦♦♦♦♦♦♦♦♦ This video was made possible by the generous support of our Patrons on Patreon! We'd like to recognize the outstanding help from our VIP Patrons: Tracy Karin Prell, Carlos Araujo, Markie Waid, Martin Stephens, David Borger, Burhan Saifaddin, Cadmus, MdeG, Michael, Richard M Haynes, Umar Khan, John Krawiec, Timothy Tapio, Samuel Ytterbrink, and Patrick Cool. Ways to support our channel: ► Join our Patreon : patreon.com/socratica ► Make a one-time PayPal donation: paypal.me/socratica ► We also accept Bitcoin @ 1EttYyGwJmpy9bLY2UcmEqMJuBfaZ1HdG9 Thank you! ♦♦♦♦♦♦♦♦♦♦ We recommend the following textbooks: Dummit & Foote, Abstract Algebra 3rd Edition amzn.to/2oOBd5S Milne, Algebra Course Notes (available free online) jmilne.org/math/CourseNotes/index.html ♦♦♦♦♦♦♦♦♦♦ Connect with us! Facebook: facebook.com/SocraticaStudios Instagram: instagram.com/SocraticaStudios Twitter: twitter.com/Socratica ♦♦♦♦♦♦♦♦♦♦ Teaching​ ​Assistant:​ ​​ ​Liliana​ ​de​ ​Castro Written​ ​&​ ​Directed​ ​by​ ​Michael​ ​Harrison Produced​ ​by​ ​Kimberly​ ​Hatch​ ​Harrison ♦♦♦♦♦♦♦♦♦♦