What is a Module? (Abstract Algebra)

[@xyzct]

I bought Pinter's _Abstract Algebra_ (Dover Books). Glancing through it, it seemed so alien. Then I binge-watched this series; it didn't take long at all. Now a quick look through Pinter looks like familiar territory. I'm 100% positive it's going to make Pinter WAY more simple and enjoyable to read for self-study. THANK YOU SO MUCH!!!

[@Socratica]

This is so lovely to hear, thank you so much!! It really inspires us to make more of these videos when we hear that we're actually helping. Good luck and let us know how you get on!! 💜🦉

[@xyzct]

@@Socratica, well the series goes well beyond just being helpful. By providing a confidence-building overview, it positions one to be able to see the _beauty_ of the subject. And being able to see the beauty inspires motivation to continue exploration. So again, my sincerest thank you!

[@richard-1604]

Snap! Pinter is very readable and this series makes it very accessible. After that I went on to Gallian.

[@ishankashyap3350]

Yess I covered Pinter halfway and then started watching these videos.... I'd say Pinter set the ground for me appreciating why certain things are included in the videos Conversely, the videos help me make sense of a lot of things that I'd read in the first half of Pinter, for example, that normal groups part with the visual image shown in the video greatly helped me understand what all of that is about.... I have finished the video series, I am sure the second half of Pinter would be a much easier read now.... These videos really helped in putting the big picture in front of my eyes and allowed me to see how different topics are related.... Loved the videos!

[@toasteduranium]

Wish there were more videos in this playlist. If I could accelerate through math at this rate in every subject, all of mathematics would be a one-year course.

[@shoopinc]

You likely won't really learn anything this way. To actually internalize things you need to be immersed in it for some time.

[@Hi_Brien]

​@shoopinc very good way to prime your kind though.

[@ImSidgr]

I love this series so much

[@terryendicott2939]

Your examples starting at 4:55 are Ideal.

[@adamtracey1848]

Yes, If A is a commutative ring, the Ideals of A are exactly the submodules of A.

[@BillShillito]

That explanation of modules was better than every single time I've tried to just read about them. The examples really helped highlight where each piece was being used (the ring and the abelian group), as well as the differences betweeh vector spaces and modules. Thank you!

[@georgebockari289]

I binged watched this series and I'm at the end...This is sad. I would love a Socratica approach to DE

[@PunmasterSTP]

It’s 3 years later, but I just did the exact same thing, and I would love a DE series as well!

[@parvadhami980]

Chal jayega

[@cmdody]

This video series are good! If i were a Abstract Algebra teacher, I would play this video in my classroom and go get some coffee while watching this with my students ;)

[@ninosawbrzostowiecki1892]

Your lectures are so awesome! I wish my professor introduced modules this way.

[@PunmasterSTP]

I’m really sad that I’m at the end of this playlist; every single video was a real (mod)jewel! Time to console myself and go back to the Python videos!

[@reidchave1441]

hahahaha "console" yourself. Nice

[@terryendicott2939]

I hope they payed you a lot of dough for that role module.

[@fernandogallardo3477]

Drum roll please!

[@tommygunhunter]

@@fernandogallardo3477 bread roll

[@santafucker1945]

wow

[@railspony]

Somewhere that bread roll turned into a cheese shtick

[@modolief]

Fascinating presentation, and very clear. I liked the examples, they were neither too advanced or too boring. I think modules are one of the lesser discussed structures in the undergraduate math classes. There are a few others about which I'd be interested to see coverage: monoids, groupoids, magmas (that's a recent one?). Also, what is meant by "an algebra"?

[@travia525]

An algebra is a vector space with a bilinear operation, e.g. Lie algebras which are also smooth manifolds

[@GoranNewsum]

We didn't do modules in my undergrad, but when I changed university to do my masters it seemed everyone on the course had done it in their undergrad; the lecturer assumed this and did a brief explanation but not one I was happy with. This video explained modules so much better than they did! Thank you!

[@Dawn_Of_AI]

I just binged 15 videos from abstract algebra. they are great!!!

[@shivamshubham7884]

the explanations are so crisp and clear.. great work.

[@akcindiamacs]

Loving the explanation in its simplest way. 🙌

[@deepalipohare1535]

Hi I am one of the follower of your channel, where are u socratica? We need next videos on Algebra as well as other subjects of maths. Please upload another videos.

[@mohammadamanalimyzada8332]

I WISH YOU ALL THE BEST. BILLIONS OF THANKS FOR THE EXTRAORDINARY PLAYLIST

[@christosgeorgiadis7462]

Great series! Thank you! Will there be one on Lie Groups?

[@alkankondo89]

Thank you SOOOOOO much for this video!! Finding a video on modules was so frustrating, since the term "module" is more commonly used to mean "a section of a course". So, it took me FOREVER to find a video on what I wanted. (I finally stumbled on this video after using the search phrase "vector space generalization".)

[@Grassmpl]

I searched "module algebra". This was the first video that came up.

[@HanyeeLim]

I wanted to take a step on abstract algebra, and I think I finally found a perfect series for me, which was such a hidden gem! Now Imma watch this whole series from now on :)

[@Socratica]

We're so glad you've found us! Good luck, and let us know how you get on! 💜🦉

[@HanyeeLim]

@@Socratica I just took the exam and it was not bad! If no Socratica, then I could've failed the test xD thank you so much!

[@Grassmpl]

Speaking of series, do a video and composition series and Jordan Holder theorem.

[@192ali1]

Where is Galois? The series and your presentations are just excellent,Thank you. You build a beautiful palace but did not finish the ceiling and the roof of that palace.What happened to Galois?? This series should end up showing that there is no general formula for polynomials of degree five and more. Am I correct? Does this series continue? Thank you

[@MuffinsAPlenty]

I wouldn't be surprised if they do some Galois videos at some point, though it might be difficult since understanding Galois theory requires a mastery of group actions and the basics of field theory - and they haven't covered these topics yet. I suppose, if done in a certain way, you can avoid group actions, but the basics of field theory are necessary. Also, sure, from a historical context, Galois theory, and in particular the Abel-Ruffini Theorem, is the crowning achievement of abstract algebra. But that's _only_ from a historical perspective. Abstract algebra is an _extremely_ useful tool a large variety of mathematics today (e.g., algebraic topology, algebraic geometry, algebraic number theory, mathematical logic, and category theory). The modern use of abstract algebra is to assign algebraic structures to other mathematical objects and use those algebraic structures to learn about the objects. In this sense, ending the series on vector spaces and modules is much more in line with how abstract algebra is used today.

[@KyPaMac]

This is really great. My introductory ring theory course didn't get as far as modules, and module theory always feels like a brick wall of abstraction when I try to read about it. Many thanks.

[@akittross]

Great job, as usual. Thank you. Looking forward to more. Maybe Group Actions?

[@Grassmpl]

Which lead to automorphisms and semidirect products.

[@josephlombardi8963]

can you please make a video about ideals? You are the only person on KZbin who makes this stuff easy to understand! Or as easy as possible!

[@Grassmpl]

Ideals are examples of modules with the standard ring multiplication.

[@yongmrchen]

Finished watching the list. Thank you.

[@Socratica]

Congratulations! You have impressive stamina. 💜🦉

[@CyrusVatankhah]

Other topics such as Topology and Algebraic Topology would make your channel much better

[@aasmaliaqat2057]

Thanks for your lecture It is very helpful to understand the modules...

[@MathsWithAsad449]

Great way of teaching pure mathematics

[@soumenghosh810]

great video! after watching this one, I decided to learn/revise modules from your channel, but unfortunate this is the last one in the series :(

[@Socratica]

Sign up to our email list to be notified when we release more Abstract Algebra content: snu.socratica.com/abstract-algebra

[@christopherkemsley4758]

Oh no! I’m at the end of this series! :( Socratica, will there be more? This has been an incredibly useful and well-made series, but there’s still so much more!

[@abdelillahjamous]

Outstanding explanation, thank you so much team Socratica.

[@thcoura]

I've made it. All videos. Thank you so much. This overview helped me a lot to get a general understanding and diminish my anxious having in hands a big book of Abstracted Algebra.

[@camerashysd7165]

yo that back up track so cool

[@random_content_generator]

You guys are awesome teachers!

[@naimulhaq9626]

Excellent !!! [Liliana, Michael and Kimberly have done a great service to Mathematics. Edward Freenkel's dream of uniting the various mathematical 'islands', seems easy for you. In your next video, I hope you can do justice to Edward's dream. ] If I could view this series 40 years ago, I probably could have the insight Witten have, enabling him to reveal the secrets of nature. But having viewed, I have a number of questions. 1) How does abstract Algebra provide insight into 'infinity' and 'zero'. What insight do we get about 'infinite sets', Cantor's cardinal/ordinal numbers, from abstract algebra. 2) What does abstract algebra tell us about 'self-reference'?

[@nandy1002]

Please also make a playlist for Linear Algebra

[@vladislavurumov538]

Incredible series, would really appreciate something about polynomials

[@sujitmohanty1]

Superb exposition....

[@Alkis05]

Nice intro to Abstract Algebra. I wish you made a series that dived deeper on specific subjects, maybe group theory. At least cover other algebraic structures that were missing, like lattices.

[@Barnardrab]

It got easier to understand when you plugged in some numbers. Additional examples would make it more clear.

[@AbdulRabChachar]

they should upload exercises.

[@YinpeiDai]

This series is amazing!!!

[@dhananjaysahani4470]

Very nice way of teaching mam

[@halilibrahimcetin9448]

Awesome video series. We are eager to gain deep insights on mathematics.So , Socratica did this job really perfect but we still have more ways to reach the destination.

[@awaisjamil6349]

Wonderful lecture series on abstract algebra.... Listened all the videos. Keep on doing such kind of work to other mathematics branches like topology, Real analysis etc

[@yonyao2950]

thnks for d basic idea ,much needed to start modules❤️

[@ayeshamuqaddas9180]

That's owesome and thanks for hard working ❤❤

[@mamadoualieujallow3091]

Thanks so much. God bless you

[@math.by.bashar]

Thanks for this information

[@anyachan567]

Great Great work.This is real education. Already shared, membership settled from today.

[@Socratica]

Thank you so much for your kind words. We're so happy you are watching! 💜🦉

[@hoolerboris]

quick question, at the definition of module. is the 1·m=m equality a requirement or its own, or doesnt it follow from the associativity (r1·r2)·m = r1·(r2·m)? we have (1·1)·m = 1·(1·m) => 1·m = 1·(1·m). I suppose that from that, we can assume that 1·m=m. (or does it require an additional cancelability thing that isnt necessarily part of the definition?)

[@MuffinsAPlenty]

1·m=m is necessary to state. For example, let your ring be Z (the ring of integers), and let M be the set of rational numbers. Define the addition on M to be the same as rational number addition. Define your scalar multiplication on M to be n·m= 0 for all n in Z and for all m in M. This satisfies all of the conditions of a module _except_ for 1·m=m.

[@musazainab7872]

Perfectly understood. Thanks so much ❤️

[@rayrocher6887]

role module funny nice lady. this encourages math.

[@worldboy9684]

Socratica, way to master a format, when I grow up I want to be Socratica!

[@MathsWithAsad449]

Nice style of teaching, i am also interested to teach mathematics like this way, can you guide please about the set up you use for making these awesome videos

[@mistertheguy3073]

I loved this, thanks!

[@Grassmpl]

Great intro. Up next time to teach cohomology theory.

[@javiermd5835]

I like to define modules in the language of actions. That is, if R is a ring and M and abelian group, M is an R-module when paired with a ring action. In the special case when R is a field, then M is an R-vector space. Modules are nothing but generalizations of vector spaces. They arise naturally when you examing ideals in a ring (that is one of the reasons why ideals are the meat and potatoes in ring theory). A nice example: every abelian group is a Z-module, it arises naturally from the endomorphism group of the abelian group structure and you get the Z-action from the canonical map from Z to any ring, it is just scaling by an integer multiple. Why are modules omitted altogether in introductory algebra courses? I don’t know to be honest.

[@sandeepk4339]

I want more videos on abstract algebra, it very helped me please👨‍💼

[@karlstroetmann1435]

I really like this series and would love to see it extended to cover Galois theory.

[@dukhtarakhtar]

I have my 1st class of this subject tomorrow nd it's seeming I'm unable to digest all this information btw you're so good in abstract algebra

[@Grassmpl]

A variety is also a class of objects in geometry and topology.

[@INAYATULLAHSHEIKH]

Outstanding series

[@sandrasurendran5068]

Want more topics in this series.

[@indubharathymurugesan1903]

Your videos are very good and helpful. Could you list the books that you are referring for algebra?

[@Socratica]

We're so glad you are enjoying our videos!! That really inspires us to make more! We're recommending the following text for Abstract Algebra right now (link below). If we come up with some more, we'll add them to the description box of the video. Good luck with your studies, and keep us posted about your progress!! Dummit & Foote, Abstract Algebra 3rd Editionamzn.to/2oOBd5S

[@KyPaMac]

A very good one, with a different feel from Dummit and Foote, is Fred Goodman's Algebra: Abstract and Concrete, which Goodman has made free of charge at homepage.divms.uiowa.edu/~goodman/algebrabook.dir/algebrabook.html. (He requests that, if you download a copy and use it, you make a donation to the charity of your choice.) Another great book with many of the same concepts is Ian Stewart's Galois Theory, Fourth Edition: www.crcpress.com/Galois-Theory-Fourth-Edition/Stewart/p/book/9781482245820.

[@gourishankarsahoo518]

Why vector space defined on a field not an integral domain

[@mariabibi1266]

simply great

[@briancrane7634]

OK by the end of your explanation my brain hurts. So I need to watch the series from the beginning. I can tell you're saying something very important and profound but I need to come up to your speed. Many thanks for your simple, clear and concise presentation!

[@Socratica]

We're so glad you've found us! Starting from the beginning of the playlist is a great idea. Feel free to post questions! We get to them when we can, and also fellow viewers often contribute great answers. Thanks for watching! :)

[@patrickyeung5771]

Thanks

[@mr.tamasamhadayatbhatti2900]

please tell me why we study vector spaces???? please clear my point

[@VarunKumar-kr3zx]

Z is module over itself but not Z is not vector space over itself and so every module need not be vector space.

[@mirceapintelie361]

oh I remember the joy of searching for the dividers of 0 in modulo n😫

[@akinwilson8799]

Thank the lord I paid attention on my undergraduate and postgraduate

[@sanatphotography6810]

Please describe about the division algebra on modules theory

[@mohamedazeem1495]

Nicely explained

[@akashgupta9962]

She gave a nice presentation.

[@markosskace514]

Next should be algebras (vector spaces with multiplication).

[@mdsaifulratul5723]

Nice....I want more simple example please...

[@javaandclanguagetutorials7721]

Thank you

[@nikitasrivastava465]

Very nice lecture...

[@saurabhsingh-ow7ue]

thank you madam....

[@defunctuserchannel]

How about a video on what an algebra is?

[@muhammadahad259]

Wowww fantastic....I like the way you are teaching.👌👌👌👍

[@amitmishra-fe6yi]

Wow really it's very helpful 🙏🏽

[@simonAdeWeerdt]

Very nice.

[@mediwise2474]

Is this playlist contains all concepts of abstract algebra?

[@abdulhameedafridi9524]

outstandig explanation..Keep it up

[@joaquimmoore2080]

Por Favor. Ponham legendas em Português.

[@Robert_ka7a]

Poetically, behold the confirmation of the ethereal journey of funds from the bank to your Visa/Mastercard card.

[@ibnezohad9666]

The great mathematician❤️👌

[@danzap3844]

"Die Module spiel´n verrückt. Mensch, ich bin total verliebt. Weil auf Liebe programmiert. Mit Gefühl. Schalt mich ein und schalt mich aus. Die Gefühle müssen raus ..." *sing

[@Sartaj122]

Nice explanation mam

[@ashishkumarupadhyay3005]

Very nice ,how can we found all guidelines which is you gives us..

[@kunslipper]

Thank you so much.

[@rajattaneja7690]

Great mam

[@jednorazowy1000]

I like it.

[@picardalrochell7580]

Module: Let's make students suffer! Double Module: That sound fun! Me: But I'm Already Suffering.. School: R U Sure about that

[@jimnewton4534]

A module is composed of a group G and a ring R. What happens (1) if I consider a subgroup of G, that does make a submodule? (2) if I consider a subring of R, does that also form a submodule?

[@MuffinsAPlenty]

Good questions! (1) Maybe. A submodule has to be a subset of G which is also a module by the same operations (the group operation of G and the scalar multiplication from R). In particular, if you have a subgroup of G, then it is a submodule of G if and only if it is closed under scalar multiplication from R. (2) No. It wouldn't be a "sub"module. However, if you have a subring of R, say S, then the R-module structure on G induces an S-module structure on G, by having the same scalar multiplication. In fact, given any R-module M and a commutative unital ring S, and a ring homomorphism φ : S → R with φ(1_S) = 1_R, then this homomorphism and R-module structure induces an S-module structure on M as well, where scalar multiplication from s is defined by s∙m := φ(s)∙m. This is called "restriction of scalars", and your question explains why one would think to name it as such: this method allows you to restrict your scalars to a subring.

[@rupertsmixtapes812]

2:26 "Module: M = any abelian group" This confused me, because modules aren't abelian groups, they're modules.

[@viktorashistan7207]

7:16 felt like an eon