Ideals (Commutative Algebra 2)
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- Опубліковано 11 жов 2024
- We'll talk about ideals with special properties and their corresponding quotient rings. In particular, we'll describe prime ideals, maximal ideals, the nilradical, and the Jacobson radical.
I was fascinated by the proof that the Nilradical is the intersection of all prime ideals, since previously I have only worked with the Nilradical by defining it as the collection of nilpotent elements in a ring. To me, this definition of Nilradical is not obvious at face value, so it was interesting to see in the proof how the concepts of nilpotence and "prime-ness" are related to one another.
Glad this gave you a new perspective!
This lecture is a nice mixture of some definitions and good proofs of some interesting results. The proof of existence of maximal ideal for a nonzero ring reminds me the proof of theorem that "every nonzero vector space has a basis" from linear algebra, by using the Zorn's lemma.
Nice comparison!
All the parts about prime ideals, maximal ideals, the nilradicals of rings, etc. reminded me of what I did last year in my independent study. For someone who is interested, prime ideals and the quotient rings by prime ideals (integral domains) play an important role in studying varieties in algebraic geometry using the language of commutative algebra.
We will have a lecture on Spec and algebraic varieties soon!
The relationships between the Jacobson Radical, units, nilpotent elements, and prime ideals are interesting. Great video. Loved the maths.
Oh Misha.
I was definitely a little confused in lecture today when you were talking about the Nilradical and Jacobson Radical since I only got to half of the lecture before class, but after watching the lecture I am able to understand what we did in class a lot more. For some of the proofs, I had to pause and read back over and flip through my notes to fully understand what some of the contexts were saying, but it made sense after for the most part. The Nilradical and Jacobson radical are actually pretty interesting to learn about and their connection to each other.
Glad this clarified things!
At 20:00, we prove that every ring contains a maximal ideal using Zorn's Lemma. Zorn's is of course equivalent to Choice, but I wonder if this is necessary. In other words, are the statements "Every ring contains a maximal ideal" and Choice equivalent? If so, I'd like to see a proof that "every ring contains a maximal ideal implies choice." I know, famously, it was a tough question to prove that choice is equivalent to "every nontrivial vector space has a basis," and took a bit of heavy machinery. The Hanh-Banach theorem from functional analysis has a similar interesting interplay going.
Excellent question! According to this 1979 paper by Hodges ----people.math.ethz.ch/~halorenz/4students/Algebra/Hodges_Krull_Zorn.pdf-- -- yes, this result (aka Krull's theorem) implies Zorn's Lemma.
@@zvirosen753 I'll take a look! Are we missing Lecture 3?
@@luna9200 Lecture 3 was in person. Hopefully I can record a video version to be posted here eventually.
(On second thought, perhaps the UA-cam comments section isn't the best place to ask logistical questions. Figured I'd ask while I checked your comment, though.)
At min: 9:12 , is it understood for this proof that the pre-image of an ideal is necessarily an ideal?
Good point! I do seem to take that for granted.