Uploaded April 2021 | Updated September 2026, 2 weeks ago
A very useful theorem in ring theory is the theorem that an ideal P is prime if and only if the quotient R/P is an integral domain (ID). Similarly, an ideal M is maximal if and only if R/M is a field. In this video, we prove both of these statements!
Ring & Module Theory playlist: youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq7
0:00 Prime ideal
3:06 Maximal ideal
9:16 Maximal implies prime
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Music: OcularNebula - The Lopez
A very useful theorem in ring theory is the theorem that an ideal P is prime if and only if the quotient R/P is an integral domain (ID). Similarly, an ideal M is maximal if and only if R/M is a field. In this video, we prove both of these statements!
Ring & Module Theory playlist: youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq7
0:00 Prime ideal
3:06 Maximal ideal
9:16 Maximal implies prime
Subscribe to see more new math videos!
Music: OcularNebula - The Lopez
![I Want to Play a Game: Proving [0,1] is Uncountable](https://i.ytimg.com/vi/Mcfnnp8rxkI/mqdefault.jpg "I Want to Play a Game: Proving [0,1] is Uncountable
The interval [0,1] is uncountable, and there are many ways to prove that. Cantors diagonal argument is a classic example. But theres another way: proof by playing a game. This video is an explanation of a proof by contradiction after assuming that the infinite set [0,1] is countable and playing a game on that interval. All we need is basic calculus for the limits of sequences!
Original based on Grossman & Turett: https://arxiv.org/pdf/math/0606253
Calculus Problems playlist: https://www.youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITao
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Music: C418 - Smooth Fall")
