Ideals in Ring Theory (Abstract Algebra)

An ideal of a ring is the similar to a normal subgroup of a group. Using an ideal, you can partition a ring into cosets, and these cosets form a new ring - a "factor ring." (Also called a "quotient ring.")
After reviewing normal subgroups, we will show you why the definition of an ideal is the simplest one that allows you to create factor rings.
As an example, we will look at an ideal of the ring Z[x], the ring of polynomials with integer coefficients.
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We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition
amzn.to/2oOBd5S
Milne, Algebra Course Notes (available free online)
www.jmilne.org/math/CourseNote...
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Teaching​ ​Assistant:​ ​​ ​Liliana​ ​de​ ​Castro
Written​ ​&​ ​Directed​ ​by​ ​Michael​ ​Harrison
Produced​ ​by​ ​Kimberly​ ​Hatch​ ​Harrison
#AbstractAlgebra #Math #Maths
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[@Socratica]

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[@Vivek-lu4eq]

What is the name of background music that you used ? Please tell me !!!! 🤗🤗🤗

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I want to thank Socratica .I'm from India🇮🇳 and by your abstract algebra video I completed my graduation which I failed last year so thank you so much❤❤❤❤😊😊😊

[@Socratica]

Congratulations!! You should be very proud about your hard work. Thank you for telling us - it really inspires us to make more videos!! 💜🦉

[@JISHUDASSAMU]

Love the way she presents concepts.

[@davethesid8960]

Well, definitely better than my teacher did.

[@Dark_Souls_3]

She’s just an actor, but I agree, the writing is amazing in these videos. Very good compared to other lectures I’ve been seeing!

[@terryendicott2939]

The comments would be Ideal if they were devoid of puns. One could however Factor out the puns and see what is Left. Would that be Right? If so then it would be Double Sided, and puns in a Class of their own.

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OMG this 12 minutes video is like a sonata, I am completely into it. Such a pleasure in my mind to enjoy mathematics. Math is beautiful, thank you.

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I wish I discovered this channel at the beginning of the semester! Great explanations!!

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Bro this is me for real

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Just loved the way how you make concepts so easy and do relate the things ❤

[@javiervega40]

Your abstract algebra videos are amazing!!! Keep making videos!!!

[@elliotf-p9722]

These videos are joy personified

[@pyrotfboba]

super happy to see these videos are being developed further!! hoping to see a video on the Sylow theorems and ring homomorphisms soon!

[@giuliocasa1304]

Thank you! This is the best explanation of ring ideals I've found!

[@samrichardson5971]

I’m sure I’d have had to see the absorption property explained like this once upon a time but this was a really neat reminder of why an ideal has to satisfy it!

[@ifyhu92]

i just finished watching all abstract algebra videos they are amazing!!! Please keep going with the content this type of learning is sooo efficient and I actually learn something

[@iliyakantoma5413]

Wow love it thank you for saving me.. I truly finds it difficult to understand this Abstract algebra before.

[@bhumikabaddhan1396]

Omg I just understood everything...you made it easy ...Thank you!

[@georgettebeulah4427]

This make so much sense and meaning and really do understand what your saying and mean. Thank you for all the video

[@jiaxuanouyang2960]

Professor didn't explain why an ideal is called ideal. Thanks a lot for clear and precise explanation!

[@Socratica]

We're so glad we could help - thanks for letting us know!! 💜🦉

[@franciscovargas2600]

These videos are incredible, would be great to see a topology series !

[@Grassmpl]

Nah they would just be homeomorphic to these videos.

[@robertkrohn9385]

These videos are incredible resources

[@xFloppyDolphinn]

These keep popping up in my recommended. I don't understand much of it but it's still pretty interesting

[@tothepoint1815]

Love the way she teaches.

[@raunitsingh676]

One word for this video: PERFECT

[@gogo-pj2lm]

Months of waiting.... Finally!!!!

[@user-we4eb6rq7j]

Thank you Michael

[@user-ho5vc5zo1x]

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[@dounia1988da]

Great job explaining

[@joetursi9573]

Wonderful presentation

[@unkown5528]

Great videos, saviour for students in the pandemic

[@jessicacarneirohatta8195]

Liliana, sou sua fã! Diva em tudo que faz!

[@osebrainquestfoundation9631]

Great concept.

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you snapped in this one 😍

[@Will-Ch]

more videos please!!. thank you so much

[@ahmadshahhimmat25]

I love your lectures

[@blessythomas7721]

I finally got what ideals mean! Can't thank you enough!!😭💕

[@Socratica]

This is so wonderful to hear! Thank you for telling us!! 💜🦉

[@NotBary]

I just passed my abstract algebra final/class because of these videos, thanks a lot. Do you all plan on making any videos covering partial differential equations?

[@Socratica]

This is so great to hear - thank you so much for sharing. It really does inspire us to make more videos when we hear they are helping! We'd love to continue our Calculus series and to also address PDEs. SO MUCH TO DO!! It's a good problem to have. 😄

[@Thats_A_Dummy_Name]

You are awesome. I would love to see a similar video series on category theory concepts.

[@Grassmpl]

The universal properties just get tensor and tensor.

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thanks a load this cleared up a lot for me

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Fantastic expositor....

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A wonderful explanation I've never seen in any text (Gallian included.)

[@lukadeclerck4702]

love your videos, just a tip: you could ask small questions to test the insight of the viewer and his understanding of what you said

[@tthtlc]

Thank you these are dream topics I have been yearning to learn since a teenager. Thanks.

[@Socratica]

We're so glad you're learning with us! We love making these videos. And thank you for so kindly supporting our channel-it means we can keep making more! 💜🦉

[@Mikey-mike]

Well done.

[@akashbouddh6056]

Thank you very much ma'am 👍👍. We are like your teaching ideas.

[@mohamedkhalid343]

you are an ideal teacher

[@iqrarali4631]

Thank you so much for such a nice lectures. Would you plz upload short lectures on finite fields, Galois theory, elliptic curves, algebraic curves

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i am early..but this video is very good... previous i had no interest in algebra..but due to you videos i like reading algebra

[@AkamiChannel]

Good stuff

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Your presentations are wonderful. I wish I had your videos when I was in college.

[@Grassmpl]

You probably had to learn off of typewriter manuscripts.

[@myworldfriends123]

@@Grassmpl No, just chalk boards. :-)

[@haris525]

Thank you! 🙏

[@doodelay]

Hey socratica, I will literally watch Liliana teach any math and science overview from real analysis to topology to electromagnetism. I do not care. This channel has helped us immensely even without a whiteboard and I fear you're not taking advantage of our admiration

[@HDQuote]

ok, so to recap (correct me, if i am wrong): assume: (G,+,·) is a ring

[@soulysouly7253]

"There are many ways you can motivate the concept of an Ideal in abstract Algebra" Ma'am YOU're motivating me to learn that concept

[@ishika01375]

what an explanation! wow

[@jeremyjakob9721]

Awesome, but I hope some more videos are coming on prime ideals, maximal ideals, principal ideals and the isomorphism theorems.

[@Grassmpl]

Also varieties and schemes.

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Me too waiting for

[@PunmasterSTP]

Best math-pun-turned-into-sponsorship-message I've ever seen...

[@ahmadshahhimmat25]

I request you to share some lectures according to Algebric topology.

[@Grassmpl]

You can learn it yourself. That's a very fundamental group. Lol.

[@snoopywlk]

I love abstract algebra because of her ideal presentation.

[@Grassmpl]

Yeah she's good at the generators and the relations.

[@mariano5704]

Thanks!

[@Socratica]

Oh my goodness, thank you for your kind contribution, Socratica Friend! It really goes a long way helping us make more of these videos. 💜🦉

[@tims2532]

Cant wait for this series to reach Galois theory!

[@charitylyngdoh8912]

Thank You Ma'am🙏

[@doodelay]

Hey socratic, I will literally watch Liliana teach any math and science overview from real analysis to topology to electromagnetism. I do not care. This channel has helped me immensely even without a whiteboard

[@swastik2484]

U teach very activel I IAM impressed

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Socratica is by far my favorite YoutTube channel so far. Ironically, I have found this channel not a week ago, but couldn't believe the quality of delivery. Specially this lady who teachs Abstract Algebra is just incredible. Hail maths!

[@OCHOA0187]

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[@usmanaziz1679]

I was extremely waiting for your videos on the topic of Abstract Algebra please upload Daily videos

[@gucker]

Daily videos? Do you think making a video takes a couple of minutes?

[@Socratica]

heehee thanks for that - if only! 💜🦉

[@usmanaziz1679]

@@Socratica welcome @Socratica 😇

[@usmanaziz1679]

@@gucker OK i know its too much difficult I'm sorry @Socratica

[@Socratica]

We're so happy you're enjoying the videos!! But yes, it does take us a lot of work. More are coming! 💜🦉

[@Ellanion]

Many of your other videos are helpful because they are relevant to me and I gain a greater understanding. This video, I barely understand a single word, but it's somehow so very meditative to listen to.

[@saurabhsingh-ow7ue]

thank you madam..............

[@TurrettiniPizza]

thanks big sis

[@sauerkirschmarmelade._.807]

what the fuck 1 minute into the video and she already made me understand factor groups and normal subgroups

[@adsoyad2607]

I've watched this video at least 8 times during the last 48 hours I'm losing it

[@Mizraab2912]

7:45 Nice!!!

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This is still here!!!!??? I haven't been on this channel in 4 years!!!

[@Mau365PP]

Thank you Michael Harrison for the lectures.

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Hello I have great appreciation and reverence for your channel and it's prominence A need you to make a Video on 1. Temperatures below absolute zero 2. Gravitational waves property. If they travel at light speed, do they have other similar properties like reflection, refraction, diffraction, doppler shift polarization. What is their wavelength range? does special relativity apply to it? 3. Collapsing an air bubble with sound underneath a liquid surface 4. Square waves

[@enpeacemusic192]

The way I see it that an ideal is generated by an element you want to act as 0. Notice that the definition of an ideal is basically the definition of 0: 0 + 0 = 0 (I is a subgroup of the additive group) and 0 * r = 0 (xr is in I for all r in the ring and x in I)

[@gaboqv]

it would be useful to add that ideals are in fact subrings if one keeps with the idea of a ring not needing an identity, I guess tough that rings with identity were more popular then too

[@Viettrantuan]

A very good expression. Could you tell me your software to make this video? Thanks so much!

[@lylechen8881]

8:18 Brilliant! 😂

[@eerisken]

Hi Socratica, thanks for the videos. But Why don't you think to prepare a MOOC about Groups & Galois Theory on a site like Coursera, or Udemy?

[@sharathkavinamoole325]

please provide full lecturing of abstract algebra.

[@chounoki]

Wait, at 8:17, earlier you said that a ring does not require a multiplicative identity "1", but now you say that since ideal may not have a multiplicative identity "1" so ideal is not a subring. You contradict yourself.

[@sharik2377]

Yes! this made me so confused, every ideal is subring but the converse isn't true.

[@latevilawson86]

Thanks for this video. I like your slide, please can indicate me the type of the beamer presentation are you using?

[@Grassmpl]

This girl's skills are next level. The math and the visuals.

[@manuelortizrobles3944]

Qué penita verte por acá, de seguro vas a triunfar como no lo hiciste en socrática en español.....

[@boradmay]

Wish my classes could be this simple and clear so I don't have to sit through 90 mins.

[@joetursi9573]

Th other five partitions of Z are the can only be wriiten compactly ,as 1.z, 2.Z, 3.Z, 4.z and 5.z ie as normal subgroups by the definition of normal subgroups of Z she just gave.

[@Leviathan-]

8:11 11:27 I think all ideals are the subrings even ideals are more than subrings because subrings contain their own element multiplication but ideal contain all multiplication of their element with any ring element. And it's not necessary for subring to contain the multiplicative identity 1 which is you telling that it is missing in an ideal then it is not subring example set of even number is the subring of the ring of the set of integer

[@usmanaziz1679]

Please make a video on 1st , 2nd and 3rd isomorphic theorem with proof also explain Homomorphisim with your concept 😭😭😭

[@Ottmar555]

There is already a video on homomorphisms.

[@howmathematicianscreatemat9226]

Would you still want that ?

[@Mrpallekuling]

Another example: The set 3Z formed by multiplying each integer by 3 forms an ideal. The quotient ring Z/3Z has three elements: 0 + 3Z = {0, ±3, ±6, ±9,…} 1 + 3Z = {…, −8, −5, −2, 1, 4, 7,…} 2 + 3Z = {…, −7, −4, −1, 2, 5, 8,…}

[@divyeshraj6306]

What is prime ideal??

[@craigwall9536]

Is there an order in which we should watch your videos in order to become familiar with these concepts? EDIT: Never mind. I found it.

[@karthikkrishnan6717]

Is ideal cardinality respects with characteristics

[@WestOfEarth]

I noticed in defining the properties of Ideals, there is the terms iy and xi -- it appears there's an order to how 'i' is multiplied -- I believe I understand this is due to Rings not being Albien under multiplication. But what does this look like beyond definitions? What would be an example? Thanks.

[@niklas6882]

First, you need to think about a Ring that ia not abelian. The easiest example should be the 2×2 matrices, e.g. in the real numbers. Then you can think about the right ideal with zeros in the Second row. If you try multiplying any matrices with other in the given ideal, you should notice that the order makes a difference 😁 Hope i was able to help & my english skills were not too bad 😂 Edit: you can get a left ideal the same way by taking the 2×2 matrices with zeros in the second column It will also work with the first row or the first column, basically works the same way 😁

[@leofun01]

11:32 - "disqualified"

[@terryendicott2939]

Serious comment. How deep are you going to go with ideals? You did a segment on modules some time ago. Please show the connection between modules and ideals. Let R be a ring with identity, 1, and let I be an ideal containing 1. What can be said about I? Are there proper ideals that may, as a set contain other ideals but are not contained in any proper ideals? That is, are there always maximal ideals? Let R be a commutative ring with identity. What do the quotient rings of maximal ideals look like? I could go on but I think that I am a bit far down the rabbit hole already. Thank for your work --- This is a great channel.

[@tracyh5751]

And also, what happens if we try to mod out by a left ideal? A right ideal? What algebraic structure do we obtain? Socratica is doing such a great job with algebra. I took algebra years ago but I still love watching these videos presented so beautifully. :)

[@terryendicott2939]

@@tracyh5751 Let A be a ring with I a left ideal. Then A/I is left A-module.

[@Grassmpl]

I think you know this stuff already. Ideals and quotient are also modules. Maximal ideal always exist due to Zorns Lemma.

[@zubairm7419]

Do you give your patreons an insight into your video production and editing process? Would love to become your patreon if you tell how you guys edit vids :)

[@Socratica]

That's not something we've done before, but we have considered creating a new channel to share with everyone the lessons we've learned on making videos and running a KZbin channel. Maybe we should do a poll to see how much interest there would be in this idea?

[@zubairm7419]

@@Socratica , that'd be great. I really just want to know about your vid creation and editing process. :)

[@fisicadubao]

Seria bom se você fizesse uma versão em português também.

[@hateranonimo1481]

Já existem Socrática português.

[@andrerangel1029]

Mas ela parou de colocar vídeos lá??? pq???

[@mikelreis3153]

@@andrerangel1029 ela é apenas uma apresentadora do canal, o dono é um cara, tem o canal em espanhol que parou faz uns 2 ou 3 anos de postar videos, faz 8 ou 6 anos que ela apresenta os 3 canais e talvez esteja esgotada e decidiram manter ela apresentando apenas esse canal, já que é o maior.