Rings (Commutative Algebra 1)

In this lecture, we'll go over the definition of rings, ideals, and related concepts.

[yooo]

Love how you used green pen for tropical semi ring

[matthewtrang]

This video did a great job by reviewing all the fundamental background on rings, ideals, properties of rings, ring homomorphisms, etc. for people who took Algebra 1 & 2 like me. I believe this video also gives people who haven’t had a chance to take any abstract algebra course a good warmup for the later sections. Besides, though I knew most of things here, I was surprised with some new examples given in this video that I have never seen before such as rng, near-ring, and especially the tropical semiring obtained when one drop the requirement that each element must have an additive inverse. Personally, I really want to see the application(s) of tropical semiring and/or tropical geometry, and (possibly) how to deal with these concepts using the language and tools of algebraic geometry and commutative algebra. Thanks for sharing your video!

[hansrajjangir]

This video reminds me the definitions of different algebraic structures but with a different approach that why do we need these properties to define the structure and what will happen if we drop some of the properties, by giving some cool examples. I, personally, was unaware about the importance of distributive properties of the ring But in this lecture Dr. Rosen nicely explained that we can derive the abelian axiom of under addition by using the distributive properties, But if we have distributive from one side only then it may not possible. That's really interesting!

[lr]

Grateful for this lecture, refreshing us on the fundamental structures we're going to be discussing/utilizing. I took my time watching and rewatching and did a little side googling myself for clarification on principle ideals and Proposition 1.2, so I have no questions, this time! (I did have two others at first but as I was writing them down, I was able to figure it out!) I do want to say I'm very grateful for the examples of each structure and how you took the time to give us ideas for the final research presentation! I, like Theresa, had never heard of those partial ring structures (sidebar: "rng" -- that's just funny) and had never taken the time, nor had I been challenged to, consider dropping properties of the structure. It is interesting to me that we can substitute the abelian additive group property for distribution on both sides and get the same result!
Thank you for this lecture, Dr. Rosen.

[theresabuscemi]

In this lecture, I was intrigued by the proof that that the commutativity of ring addition follows from other ring axioms - specifically, the presence of multiplicative identity and distributive properties - and does not need to be stated separately. From this discussion, we saw that eliminating certain ring axioms will lead to different structures, such as semi-rings, rngs, and near-rings which are terms I was previously unfamiliar with. I am interested to learn more about these structures, especially the given example of the tropical semiring.

[amrita]

This lecture was very beneficial in reminding me of the rules and definitions I learned in Modern Algebra. It lays a solid foundation to what we will be building upon this semester. I still have some trouble understanding kernals and proposition 1.2 #3. I had never heard of continuous function rings before until we went over them in class. I found (ag)(bh)=(ab)(gh) interesting with (ab) being multiplication in the ring and (gh) being the group operation.

[luna]

While this will 100% change throughout the semester, I do have to admit that there are not any questions I have on this material. It serves a different purpose than the rest of the lectures will. This has been an excellent review of many ring-theoretic concepts learned in Abstract Algebra 2, which of course is going to be very beneficial for this course.

[mishaklopukh]

Interesting lecture. I am having trouble figuring out why rule 3 of ring homomorphisms, f(1_A)=1_B [17:15], would be necessary. I understand that the proof that f(0_A)=0_B [19:28] would not necessarily work for 1 as you may not be able to cancel with inverses. However, it still shows that f(1_A) acts as a multiplicative identity for the entire image subring.

[ycchen]

Dear Professor Zvi Rosen:
Thank you so much for sharing such a great lecture. Really appreciate it.
Also may I ask that will you cover all the materials from Macdonal and Atiyah's books? I am very much looking forward to your lecures on the last two chapters.
Thank you so much again!

[ycchen]

These lectures are great! Really appreciate it.
And also I would like to share a math study message here for anyone who might be interested: I would like to read <Presentations of Groups> by D. L. Johnson. And I would like to ask if anyone is interested to read together? Because having a reading partner will make study more fun, especially when go through detail proof together. You are also welcome to share this message to anyone who is interested. Thank you so much!