Ideals in Ring Theory (Abstract Algebra)

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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
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Milne, Algebra Course Notes (available free online)
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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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Опубликовано:

30 сен 2024

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Комментарии : 213

@Socratica 2 года назад

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

@Vivek-lu4eq 3 месяца назад

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

@terryendicott2939 4 года назад

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.

@DejiAdegbite 4 года назад

Niiice.... ;-)

@akashbiswas4666 3 года назад

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 3 года назад

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

@chounoki 3 года назад

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 3 года назад

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

@JISHUDASSAMU 4 года назад

Love the way she presents concepts.

@davethesid8960 Год назад

Well, definitely better than my teacher did.

@Dark_Souls_3 11 месяцев назад

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!

@NotBary 3 года назад

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 3 года назад

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. 😄

@TheDopplerEffect_ 3 года назад

I don't usually comment on RU-vid videos, but damn this channel is the only thing helping me pass my third year abstract math class and I am so thankful that it exists. A sincere thank you from South Africa!!!

@larsmees4952 4 года назад

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

@tiagonata1734 10 месяцев назад

Bro this is me for real

@junchichu8703 4 года назад

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.

@ahmadshahhimmat25 4 года назад

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

@Grassmpl 2 года назад

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

@pupnoomann7866 4 года назад

for r in r and i in i i r and r i r in i :)

@Socratica 4 года назад

We all kept laughing when trying to record that!

@Leviathan- 2 года назад

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

@doodelay 4 года назад

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

@jeremyjakob9721 4 года назад

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

@Grassmpl 2 года назад

Also varieties and schemes.

@bakkamydestination6275 2 года назад

Me too waiting for

@usmanaziz1679 4 года назад

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

@Ottmar555 4 года назад

There is already a video on homomorphisms.

@howmathematicianscreatemat9226 4 года назад

Would you still want that ?

@Atul_Thakur97 4 года назад

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

@dayvidson464 4 года назад

FAZ VÍDEOS NO CANAL DO BRASILLLLLLL!!!!!!

@jiaxuanouyang2960 3 года назад

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

@Socratica 3 года назад

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

@xFloppyDolphinn 4 года назад

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

@iliyakantoma5413 4 года назад

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

@jessicacarneirohatta8195 4 года назад

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

@ggg148g 4 года назад

Amazing, wonderful, the clearest and most useful explanation ever. THANKS!!! Giving some money right now!!!

@bhumikabaddhan1396 4 года назад

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

@HDQuote 4 года назад

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

@eletronica_do_airton 4 года назад

Seria bom se você fizesse uma versão em português também.

@hateranonimo1481 4 года назад

Já existem Socrática português.

@andrerangel1029 4 года назад

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

@mikelreis3153 4 года назад

@@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.

@craigwall9536 4 года назад

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.

@divyeshraj6306 4 года назад

What is prime ideal??

@boradmay 4 года назад

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

@Mrpallekuling Год назад

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,…}

@usmanaziz1679 4 года назад

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

@gucker 4 года назад

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

@Socratica 4 года назад

heehee thanks for that - if only! 💜🦉

@usmanaziz1679 4 года назад

@@Socratica welcome @Socratica 😇

@usmanaziz1679 4 года назад

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

@Socratica 4 года назад

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

@blessythomas7721 2 года назад

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

@Socratica 2 года назад

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

@soulysouly7253 2 года назад

"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

@BrynSCat 3 года назад

This may sound strange but is abstract algebra actually complete ?.If You treat a every number as already being 4d with 360 twist.(Hopf Vibration)Euler's identity is a realistic equation for a 1D number line. In Hopf Fib ration A Ring will always be a Field relative to another number ?.

@joetursi9573 2 года назад

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.

@ياصاحبالعصروالزمانأغثني 4 года назад

عاشت ايدك على هذا الشرح الاكثر من رائع💙

@enpeacemusic192 10 месяцев назад

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)

@lukadeclerck4702 4 года назад

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

@danielcanas1802 4 года назад

Hola BONITA revisa el teorema de TALES en uno de tus vídeos esta equivocado. T d TALES Dice: Si dos rectas no paralelas que se cortan por un sistema de rectas paralelas. Los segmentos resultantes de una de las dos son proporcinales a los segmentos obtenidos de la otra. Corregir es mejor que ..... Saludos dede California.

@jorgemiltonchavarromoreno2632 4 года назад

El universo es un lugar bastante grande, lo mismo que el botón para suscribirte. No te voy a decir que hagas click en él por que estoy segura de que tú harás lo correcto... Y lo correcto es hacer click en el botón. The universe is a pretty big place, as is the button to subscribe. I am not going to tell you to click on it because I am sure that you will do the right thing ... And the right thing is to click on the button.

@theuberman7170 4 года назад

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

@terryendicott2939 4 года назад

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 4 года назад

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 4 года назад

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

@Grassmpl 2 года назад

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

@JoSh-yu6jt 4 года назад

This.Series.Is.Genius !!!! A big thanks to all the patreons. Such educational videos are a gift. 💯

@davidmilan2846 Год назад

I thought that multiplicative identity wasnt a requirement for a ring. Why is it a requirement for an ideal to be considered a subring then?

@gavinsuddarth696 4 года назад

Yeah no. I don't get this all math. Just something beyond me.

@sauerkirschmarmelade._.807 2 года назад

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

@franciscovargas2600 4 года назад

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

@Grassmpl 2 года назад

Nah they would just be homeomorphic to these videos.

@crartefatosdecimento2299 3 года назад

Eu te conheço. Tu fez novela no Brasil. Tell me. Are you american or Canadian?

@مريمعزيزجليلمهديالخزرجي Год назад

What is the solution ideal Z15 Please reply because I have exam 😭

@harrykainth788 8 месяцев назад

Mam Give me your brain for just today, that i can able to give my exam 😂❤

@javiervega40 4 года назад

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

@adsoyad2607 Год назад

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

@AndreLemus 4 года назад

Es mejor en el español, si hubieras seguido tendrías más gente 👌🏻 Más visitas y suscriptores.

@Will-Ch 11 месяцев назад

more videos please!!. thank you so much

@samrichardson5971 4 года назад

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!

@ayaka.11 8 месяцев назад

"Ideals are not technically subrings" All ideals are subrings by definition, right? hmmm I am confused

@MuffinsAPlenty 8 месяцев назад

It depends on the definition of subring you use. In the past, many algebraists did not require the existence of a multiplicative identity as part of the definition of a ring. But more and more today, algebraists think that the existence of a multiplicative identity _should_ be part of the definition of ring. So in order to be a subring, an ideal should have a multiplicative identity, then, and even more, should have the _same_ multiplicative identity as the ring itself. This means that the only ideal which could ever be a subring is the unit ideal - the full ring. But that older definition where rings don't have to have a multiplicative identity, yeah, that allows all ideals to be considered subrings.

@prashantanand6552 2 года назад

I got a bit confused. The video says that ideals are not a subring but the Gallian book says ideals are subring. Actually their definition starts from the statement that " A subring A of a ring R is called an ideal if....."

@Yougottacryforthis Год назад

The identity of multiplication isnt always necessary to be a sub-ring, that varies between professors. If you accept it doesn't require the identity then an ideal is for all intents and purposes the normal sub-ring.

@TurrettiniPizza 4 года назад

thanks big sis

@GandharKulkarni2000 4 года назад

You're too late Liliana-- I'm already done with my algebra course

@LeandroSantos-yd3nj 4 года назад

well she isn't for freshmen...

@louisl7346 Год назад

10:59 Why are ideals not called normal subrings

@doodelay 4 года назад

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

@OmarAhmed-v7v 9 месяцев назад

Hello, can I communicate with this teacher?

@GopherManlet 4 года назад

Huh I'm early

@rbkstudios2923 4 года назад

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

@jorgemiltonchavarromoreno2632 4 года назад

The universe is a pretty big place, and so is that subscribe button. I´m not going to tell you to click it, because I´m certain you´ll do the the right thing... the right thing is to click the button. El universo es un lugar bastante grande, y también lo es ese botón de suscripción. No voy a decirle que haga clic en él, porque estoy seguro de que hará lo correcto ... lo correcto es hacer clic en el botón.

@joetursi9573 2 года назад

I can't see why she says 'consider the normal subroup 6Z. This normal subgroup partitions the integers into cosets. But only 6Z,the normal subgroup is a group. Any of the five other cosets are simply sets. That not true since she just said the subgrops are nZ where n is in Z Comments please.

@MuffinsAPlenty 2 года назад

The cosets of 6Z in Z are 6Z, 1+6Z, 2+6Z, 3+6Z, 4+6Z, and 5+6Z. Of these cosets, only 6Z is a subgroup.

@danielmago4327 3 года назад

what's the name of this professor?, she's very good

@sebastianhilscher8072 4 года назад

Whether or not ideals are subrings, at the end you mentioned them not being subrings, does in fact depend on your definition of a ring. If you require it to have a multiplicative identity, it is of course almost never a subring. The usefullness of defining a ring without multiplicative identity lies exactly herein: most concepts proven for rings do ac generalize to ideals, as you do not always need the property of the multiplicative identity in the proof. So why not call them rings then?

@coolankush100 4 года назад

Finally, someone has said it. This is exactly what I've been thinking. They should comment or add a typo section or explanation in the description.

@jthomas3584 4 года назад

@@coolankush100 It's not a typo, explanation would be the right word. For the most part people define rings such that they have to have a multiplicative identity.

@coolankush100 4 года назад

@@jthomas3584 thankyou for such clarification. It'd have been nice if they explained that.

@tracyh5751 4 года назад

One reason to not call them rings is that much of ring theory uses the existence of maximal ideals. These are not guaranteed to exist when the ring does not have an identity element.

@MuffinsAPlenty 4 года назад

One reason _not_ to do that is module theory. The concept of a vector space is one where a field of scalars is allowed to act on an Abelian group. A module is a generalization of a vector space, where the scalars only have to form a ring, not a field. Module theory is an incredibly rich theory, which is useful in many branches of mathematics and can provide deep insight into rings themselves. An ideal in a ring R is always a module over R. Subrings of R are almost never modules over R. Many of the intuitive results for modules are completely false for subrings. For example, if N is a submodule of M, then the (Krull) dimension of N is less than or equal to the dimension of M. However, if S is a subring of R, it's entirely possible for the dimension of S to be larger than the dimension of R.

@littlenarwhal3914 4 года назад

Weird how you straight up disqualify ideals as subrings because they lack multiplicative identities while in your definition of rings you say that the definition is wavy and rings can either have or not have a multiplicative identity

@eerisken 4 года назад

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?

@jadtawil6143 3 года назад

i dont understand why (x + I)*(y+I) = x*y + I

@luan_ferreira_ 4 года назад

Que inglês bonito

@epuremath 3 года назад

Why you require subrings to have 1? It's wrong. All ideals - are subrings.

@MuffinsAPlenty 3 года назад

"Why you require subrings to have 1? It's wrong." No, it isn't wrong. There are two commonly used definition of "ring" which don't agree. Requiring 1 is the more modern definition of ring. In such a setting, ideals are almost never subrings. You are using an older definition of ring.

@tthtlc Год назад

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

@Socratica Год назад

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! 💜🦉

@karthikkrishnan6717 2 года назад

Is ideal cardinality respects with characteristics

@malakian6668 3 года назад

Vim ver você aqui, pq sumiu do canal Socrática do Brasil?!

@ifyhu92 5 месяцев назад

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

@hernan.herrera0 4 года назад

vendo del canal de socrática en español que lo descubrí hoy viendo los videos de las leyes de kepler

@tothepoint1815 3 года назад

Love the way she teaches.

@Nayanexx 2 месяца назад

She was an amazing actress in Brazil.

@bayupatten4777 6 месяцев назад

thanks a load this cleared up a lot for me

@manuelortizrobles3944 4 года назад

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

@majestic7768 4 года назад

A lot of subatomic geometry was involved in bringing this person into being.

@PunmasterSTP 3 года назад

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

@gogo-pj2lm 4 года назад

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

@joetursi9573 Год назад

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

@mariano5704 3 года назад

Thanks!

@Socratica 3 года назад

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

@live2laugh2 3 года назад

Come here to find definition but got whole think clear really ❤️❤️❤️

@zubairm7419 4 года назад

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 4 года назад

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 RU-vid channel. Maybe we should do a poll to see how much interest there would be in this idea?

@zubairm7419 4 года назад

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

@OCHOA0187 4 года назад

Hola hace poco vi uno de tus videos sobre las estrellas de neutrones, pero lo vi en un canal tuyo en español lo abandonaste hace 5 años cosa que no note hasta que me suscribi, quise ver mas de tus videos en este canal en español y me encontre con el video de un anuncio en el cual indicabas que volverias, creo animarte a que vuelvas en especial con los temas de astronomia y fisica, veo que aqui en tu canal de español no te ha ido muy bien que digamos pero he visto otros canales en español que les a ido muy bien y han crecido mucho hasta tener millones de suscriptores, te animo a que cumplas con lo dicho y vuelvas a tu canal en español saludos desde colombia.

@NancyYoung-r9b 18 дней назад

7715 Beatty Light

@mayank.justmayank 2 года назад

people just hit the button.

@stevehan3498 2 года назад

The only ideal I see here is you

@gaboqv 4 года назад

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