Ring Definition (expanded) - Abstract Algebra

[@Socratica]

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[@TinyMaths]

Three hours of head-scratching while staring blankly at my lecture handout. Four and a half minutes into this video, complete clarity.. Thank you.

[@deshantdevkota2563]

He prolly taught you some more things.

[@endingalaporte]

here lies the future of teaching

[@weirdnameucantthinkof4179]

Same I just watched this and I study in french still understood better then a whole 3 hrs lecture

[@user-ak7as2hi9q]

Same

[@ravikantpatil3398]

True

[@Uuuuhhhhhh]

"Consider these four sets" Me: "okay" "The integers.." Me: "Mhm" "The real numbers.." Me: "Yeah" "2-by-3 matricies with real entries and polynomialswithcomplexcoefficients" Me: ఠ_ఠ

[@stevenmann5857]

Hi, My name is Bernie sanders and I am once again asking you to explain what an isomorphism is

[@meenakshi6452]

🤣🤣🤣lol

[@yashkaliapiano]

lmao

[@MrOvipare]

Indeed this is going to make a lot more sens to you if you already have a undergrad math/physics background hehe.

[@artichokeheartbreak2279]

😂 Don't worry, you're doing better than me

[@escobasingracia962]

You can't imagine how much I was waiting this series to continue. Thanks Socratica

[@sterlingveil]

How am I only now discovering this channel in 2019? Oh how the KZbin algorithm scorns me! Instantly subbed.

[@Naatak1]

I found in 2021 what should I do now?

[@a-keyspiano9399]

I found it last month. How various channels have abused me. 😢

[@artichokeheartbreak2279]

@@Naatak1 Me too. It's perfect to watch when I want to pretend I understand complex mathematics.

[@kambouziab]

Found it in 2022…

[@ngomusoqwabe4684]

Me too

[@oppproto]

I’m studying immunology so I don’t know what I’m doing on an abstract algebra video but it’s great

[@stumbling]

Immunizing yourself from over-specialisation?

[@navjotsingh2251]

Groups are used in biology and chemistry. Maybe that'll apply to you in some way.

[@thabolikotsi5765]

I am studying Mathematics and Applied Mathematics this is my home

[@memesupreme905]

Procrastinating

[@mrdominguez3881]

@@thabolikotsi5765 x2

[@timhourigan6257]

I can't thank you enough, Socratica! I can now make a concise taxonomy/categorization of all of those objects we find in abstract algebra. An added bonus: I can now use and "translate" the textbooks I bought all those years ago that my professors never used during the course. Huzzah!!

[@RahulChauhan-zz1nt]

Great explanation. Love ❤from 🇮🇳India

[@mahimakalani2647]

Thank you Mam and you entire team for helping us understand the intuition of these concepts of ABSTRACT Algebra :)

[@onkarkalpavriksha8676]

THIS IS WHAT I CALL A CLEAR EXPLANATION !!!

[@drewtmacha34]

Please make a linear algebra series! I really enjoy the abstract algebra playlist from Socratica. I recently took an online class in abstract algebra, and I didn't fully understand the concepts until I discovered this playlist. The teaching and animations make it much more accessible than reading from a book with no illustrations (as was my case). I also took a graduate-level linear algebra class online, and, well--same thing. We read from a textbook without any illustrations, and I didn't understand much. I remember the class was mostly structured to work proofs here and there building up concepts until we developed the Jordan normal form of a matrix, and then after that I was especially lost, to where I can't even remember what topics were taught. I would be extremely interested in a playlist of linear algebra videos covering those sorts of topics.

[@starbooi4699]

Please do a series on complex analysis (final year, undergraduate).

[@hopy51]

How much can you exlain about complex analysis in such short videos though? The thing about abstract algebra is that you can explain a lot about them by simply giving definitions and examples. In complex analysis it is very different-there is the definition of a holomorphic function, but after that it is all about theorems and proofs. Really hard to give an intuition.

[@navjotsingh2251]

@@hopy51 tbh it can be done. I've seen lots of math websites present complex analysis in a very intuitive way, on par with this woman in the video.

[@Grassmpl]

You can present the residue theorem

[@arjunroyihrpa]

Really liked the way she explained so easily yet with such clarity.

[@irynado6548]

Omg... 7 yrs ago I was looking for this everywhere. This and partial differential equations.

[@k0185123]

excellent video! I'm an electrical engineering graduate student who was major in physics. Your explanation is so clear. It reminds me of my memory when I was a college student

[@mi9807]

The presentation was awesome. Its just groups but the background music made me feel like I'm unravelling the secrets of the universe

[@valor36az]

I learned so much from this one video

[@fatemekashkouie3662]

The most beautiful explanation for Algebra.

[@doodelay]

For those learning this in the future and reading this comment, remember that these definitions are almost entirely based on a single concept and that is the notion of "closure." If you understand closure and see why you need it to even sensibly discuss operations between elements in a set, then you'll see where these definitions of rings, fields, and other abstract algebra definitions arise. Without a notion of closure, every time you perform an operation in a set A that you prefer, you may find that the given operation, O, yields an element not in your set. So, at that point you say that A is not closed under O. So, you don't do O when studying A and this will allow u to strictly remain in A and study it freely.

[@mpaulshore]

You can feel free to reject the unwanted operation: it won’t O you anything.

[@mpaulshore]

Of course you might find yourself haunted by the thought that you’ve missed out on a really great O.

[@sunilrampuria7906]

A simple example of a ring without identity is the set of even integers. Division rings are also called skew field. A algebraic structure which has all the properties of a group except that of an inverse is called a monoid. So a ring (with identity) is a group under addition and a monoid under multiplication.

[@EnginAtik]

This is the best explanation of Rings ever anywhere.

[@PunmasterSTP]

That word-scramble music was pretty intense! But I like the intense bits of music; it takes me by surprise and keeps me engaged. Thank you so much for these videos!

[@gulcedemirel9820]

You are the best! I suddenly started to be interested in abstract algebra. Thanks a lot.

[@nickhodgskin]

Thank you much!! My head has been swimming with all these different words and structures, but you've helped to make it succint for me, and easy to visualise!! :D

[@Socratica]

This is wonderful to hear. We're so glad you're finding our videos helpful!! 💜🦉

[@kushal7406]

Superb, mam and socratica team, You made this concept much much much easier, Thank you very much.

[@MuffinsAPlenty]

There are uses for rings without a multiplicative identity, but I'm in the camp that believes the official definition of a ring should include a multiplicative identity. This comes from the convention of the empty product - a product with no factors. Technically, multiplication in a ring is a binary operation, so it has exactly two inputs: no more and no less. Now, there is a very natural way to extend the definition of multiplication in a ring to take on more than two inputs, and that is to do the multiplication recursively. In other words, a·b·c is defined to be (a·b)·c, and a·b·c·d is defined to be ((a·b)·c)·d. But there are, a priori, many ways one could define multiplication for more than two inputs. For example, we could have chosen to define a·b·c to be a·(b·c) or to define a·b·c·d to be a·(b·(c·d)) or (a·b)·(c·d) or (a·(b·c))·d or a·((b·c)·d). Luckily, rings require multiplication to be associative, so the associative property of multiplication allows us to extend from two factors to more than two factors without worry about making a choice in these definitions (since all possible choices end up being equal). So this is all nice for products of two or more elements, but it doesn't allow us to say that we can take a product of any finite number of elements. In order to do that, we have to be able to take a product with one factor or even zero factors. These situations can sometimes be useful! For example, if you want to take the product of all elements in a ring satisfying a certain property, then you need these sorts of notions if your ring only has one element satisfying the property or if your ring has no elements satisfying the property at all. Based on using the associative property to extend multiplication to more than two factors, we can again turn to the associative property for extending multiplication to fewer than two factors. If we demand that a product with one factor is consistent with the general associative property, we must conclude that the product of one element is that very element. Similarly, if we demand that a product with zero factors is consistent with the general associative property, we must conclude that the product of zero elements is a multiplicative identity. So if we want to be able to take a product of any finite amount of elements in a ring, we must have a multiplicative identity in the ring. This mindset also explains why many texts which require rings to have a multiplicative identity then go on to require that a subring must have the same multiplicative identity as the ring itself and why ring homomorphisms must send the multiplicative identity to the multiplicative identity. All computations in a subring should be consistent with the same computations in the ring itself, so the empty products should yield the same results (giving that we should have the same multiplicative identity). Moreover, homomorphisms are multiplicative functions and as such should send the empty product to the empty product (giving that the multiplicative identity should be mapped to the multiplicative identity). Again, this is not to say that non-unital rings or non-unital ring homomorphisms are unimportant. But philosophically, it seems to me as though they should be viewed as lacking a little bit of the structure that a ring "should" have (empty products).

[@juanpablocordon4776]

Please please PLEASE continue this series

[@MichaelGoldenberg]

Very clear coverage of basic definitions with some of the rationales for the different structures.

[@whatever5867]

This helped me a lot to understand deeply

[@user-tb5io7xt9w]

Thank you so much! It really helped to understand the idea

[@gaaraofddarkness]

I love your explanation maam. Thank you soo much.

[@MrWandalen]

If I understand correctly, mistake in the video at 2:19. Polynomial with complex coefficients has division which always keep it in the set, so should be check ( not cross ) at 2:19. Fraction complex coefficients is convertable to polynomial with complex coefficients by multiplication by denominator. Zero in denominator is only exception.

[@MuffinsAPlenty]

No, you are mistaken. The quotient of two polynomials is not guaranteed to be a polynomial. The coefficients are pretty much irrelevant because it's the "polynomial" part that messes up the ability to divide. For example, "1" is a polynomial (seems silly to call it that, but it is), and "x" is a polynomial. But 1/x is not a polynomial. Rather, 1/x is called a "rational function".

[@sobhinthomas5048]

This channel really makes maths so popular.About half a million views for abstract algebra classes woww...

[@Socratica]

Math Club!!! 💜🦉

[@kanuos]

Having a test on the weekend helped me find some really cool math channels. Thanks for the amazing explanation! Subbed!

[@theboombody]

Great explanation about the benefits of translating our thoughts more to inverses than directly defined opposite operations.

[@mohammadamanalimyzada8332]

well explained. thank you a billion times. i wish you could continue this playlist. the king of all playlists i have ever come across so far

[@rogerwilcoshirley2270]

Excellent clear and concise introductions to basic topics in abstract algebra which has now become one of my top priority projects.

[@Socratica]

We're so glad you've found us! 💜🦉

[@passhabi]

Your tone of voice and the atmosphere of the videos is like in video games that give me instructions for an entertaining battle.

[@zhenma8053]

I love this channel! I wish it existed when I still was a student.

[@djeovs]

This is what I call the art of defining! Would love to see a Socratica definition for Function Fields

[@bayupatten4777]

doing a level further maths right now and it felt like i finally have learned enough to comprehend videos like this then you said "quirky four dimensional abstract" ....

[@j4k3j0n3s]

Thank you for creating this series. If this had been available when I was in college my grades would have been so much higher. Your students are fortunate!

[@nitin6801]

One of the best channel for mathematics lovers ❤️

[@elmarose3212]

Thank you so much ! This has given me a proper idea of what a ring actually is !:)))

[@AlanTuringWannabe]

This was very clear.

[@souravacharya]

The way you teach such topics is simply outstanding.... 👌

[@estabelmk5506]

Hey, I'm really impressed with your videos. Thanks😘

[@fminc]

Very clearly and elegantly presented.

[@Jabber_Wock]

Great to see this series is live again! Thank you for your excellent tutorials, I’ve found them to be very helpful and informative. Hope to see many more in this series.

[@haidermehar2422]

A great video on this topic .Great work. I highly recommend this.❤

[@KennethMe]

Where was this when I was taking modern algebra? Great channel great videos. Keep up this great work.

[@Vindignatio]

this channel kicks ass, and so do you.

[@cheezu1265]

thank you so much for this. currently studying for finals for algebra and i cant wrap my head around these abstract concepts until i saw this :)

[@Socratica]

That's wonderful to hear!! Good luck - we're rooting for you!! 💜🦉

[@ifathameedshora7063]

Words are not enough for this amazing amazing video

[@tonym5857]

Really nice video, remember me my University math classes but most outstanding is her I ve watching videos speaking English, spannish and portuguese ... Girls on Power 🙌🙌🙌

[@SIGSEGV1337]

I just started watching so I have no idea what half this stuff means but I am very much enjoying myself

[@antho8070]

All those videos about algebra are amazing, I'm loving them ! Thanks a lot from France !

[@knaxel]

the writer is a legend thank you Michael​ ​Harrison, thank you ​Liliana​ ​de​ ​Castro for working with them, and thank you Kimberly​ ​Hatch​ ​Harrison for making this freely available here

[@Charles-xc6sr]

Thank you, you are wonderful.

[@ceasarnjuguna5954]

Awesome. Good job

[@kishanprakash007]

Brilliantly explained. So all fields are rings because fields are rings with extra steps (commutative and with identity for multiplication).

[@nileshbagada1729]

Hey ma'am please make more videos like this

[@Grassmpl]

Talk about projective space and Grassmannians. I'm confused with the exterior products of a vector space.

[@UnskilledShots]

I’m a big fan..... you guys are doing it great....

[@yahya5308]

GUYS U ARE MAKING A GREAT A GREAT SERIE . PLEASE MAKE IT EVERY WEEK

[@benjaminmcgahee5934]

What happens in the ring, stays in the ring. True for algebra and in boxing!

[@proggenius2024]

"If a division ring is commutative, then we call it a ring", what a beautiful ending sentence! Loved the way you teach and your intellect!

[@RobertLock1978]

Very clear and concise explanation.. :))

[@frankfrancia2053]

Fantastic !!

[@fionamcintosh1523]

Thank you so much for these videos

[@meenakshi6452]

You are a ginnie for us...how can something that seem so complex can be broken down so simpler and easy....

[@shreyanmullick4102]

Beautifully explained ❤️❤️❤️ many doubts got cleared thankyou ☺️

[@swatishirsath5155]

Hi, I really like the way you explain abstract algebra. can you conduct a Guest lecture for my students?

[@highermathematics-bx4mi]

Assalam-o-alaikum jazak Allah mam Allah shower his countless blessings upon you ameeeeeen Allah succeeded us In every field of life ameeeeeen

[@naksatrabailung1520]

Was studying with your playlist Just Today.. Thanks for the explanation

[@user-wn1dd8ls2u]

5:06 I often hear a concept of non-associative ring, such as 3-dimensional vectors with vector product, and other Lie rings

[@ivanvoid4910]

Why it's so transcendental? Like it so much!

[@castohernandez8236]

Thanks for a clear and brilliant explanation. Your videos help people. Please keep it up!

[@AkamiChannel]

This connected a lot of dots for me. Thank you.

[@AlessandroZir]

thanks!! very clear; ❤️❤️

[@123TeeMee]

I wish I hadn't just stared at wikipedia pages and given up when trying to understand abstract maths stuff before, because stuff makes a lot of sense now. I got back into the rabbit hole wondering what set of operations I could use for defining cellular automata rules

[@Socratica]

We're so glad you've found us, and things are becoming more clear now! 💜🦉

[@random_content_generator]

thank you for existing \o/

[@samuelokon8842]

Nice Explanations

[@mdmahmudhasan1645]

Nice presentation.

[@terryendicott2939]

I was waiting for a bad pun at the end, alas no joy. I still gave you an up-thumb. On a serious note, most introductions to algebra start out with group theory. However I think that there is a distinct advantage with starting out with rings. Groups have the advantage of only one operation, but to get good examples of groups one has to get fairly abstract, like symmetric groups, alternating groups, dihedral groups etc. Where as in your introduction you introduced rings with identity, non-commutative rings, and fields with out breaking a sweat. One can get to rather interesting types of rings with examples from high school Examples: Integral Domains (Z, Q, R, C, Z[x],....), rings with zero divisors (integers mod 6, 2x2 matrices over Z, ..) PID and UFD (Z, R[x], C[x]), Noetherian but not PID (Z[x], R[x,y]).

[@MuffinsAPlenty]

I've seen some professors agree with the approach of starting with rings since it is more intuitive. I haven't decided what I will one day do when I have to teach abstract algebra. But it's a serious consideration that people should take.

[@MrWabouz]

Both clearly explained and exact. What is your curriculum in Maths?

[@korawichbikedashcam6293]

Didn't search for this but glad to be here 😅

[@Usrbinry4n]

Thank you for this video!!!

[@nebulouskoala9058]

Really clearly laid out! Thanks! :)

[@ammyvl1]

i understood a group i understood a field you got me with the ring

[@warrengibson7898]

The name “ring” calls to mind a certain geometrical shape. It would help to know why the name was chosen for this kind of group and whether there is a geometric analogy that I am missing.

[@niklas6882]

The name historically is from germany (richard dedekind, david hilbert), but today, ring (in german also ring) most times has the same meaning as in english. But in german, you can use the word ring in a different context - it means that some things (e.g. persons) form a bigger thing. For example the word "verbrecherring" (english: ring of criminals) means that some criminals work together to reach certain goals. However, this meaning of the word ring is not used that much in germany anymore. So, the answer to your question sadly is no, it only comes from the other meaning that the word has in german. Hope my answer wasnt too confusing and my english well enough to explain it 😀

[@warrengibson7898]

@@niklas6882 Vielen dank. Sie schreiben englisch sehr gut

[@niklas6882]

@@warrengibson7898 oh, hätte ich das mal vorher gewusst 😂 aber gut, dann scheint sich die frage geklärt zu haben 👍😀

[@turokg1578]

@@niklas6882 bro thanks a lot... ive been thinking about why they call this shit a ring for a year now

[@santhoshwagle9857]

you deserve more subscribers and more likes

[@suklocharankisku116]

Almost three years after seeing you ..feeling Good.

[@filipve73]

- Abstract - @1:27 The table is written out in terms of "Absolute set of operations (+,-,x,:) " Let suspose (for the sake of argument) you can the divide by zero then the Ring would split in two different entities parallel and anti-parallel to each other. Should then speak in terms of anti-additive and anti-multiplicative properties for that Ring ??

[@MrRonluker]

I have no idea how I would apply this, but cool enough explanation.

[@brbhati3172]

Great learning experience!! 👌👌👍🖋📓

[@user-xs9oo9gc7u]

A bit of missing part: multiplicative inverses are only applied for none zero (i.e. additive identity) elements. Obviously 1/0 is not defined.

[@ishanmaheshwari5751]

Lebesgue integration... Pls pls pls pls pls ....mam upload these...🙏🙏🙏🙏🙏

[@csanadtemesvari9251]

lol

[@Grassmpl]

There are plenty of measure theory content out there.

[@dsutsav9579]

i am from india . i see your all video . you are unque in youtube space . love you mam

[@FarooqAhmadiub]

Helpful Thanks a lot