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

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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[akashbiswas]

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❤❤❤❤😊😊😊

[TheDopplerEffect_]

I don't usually comment on YouTube 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!!!

[larsmees]

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

[junchichu]

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.

[terryendicott]

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.

[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?

[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

[jiaxuanouyang]

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

[ifyhu]

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

[JoSh-yujt]

This.Series.Is.Genius !!!!

A big thanks to all the patreons. Such educational videos are a gift. 💯

[HDQuote]

ok, so to recap (correct me, if i am wrong):
assume:
(G, +, ·) is a ring <= (G, +) is a kommutative/abelian group; (G, ·) is a monoid; (+, ·) are distributive
(H, +, ·) is a ring <= analogue to (G, +, ·)
then H is an ideal of G iff (if and only if):
H ≤ G <= H is a normal subgroup of G <=> ∀h ∈ H, h+G = G+h <=> You can construct (G, +)/(H, +) and it is a (quotient)group
∀h ∈ H, ∀g ∈ G : h·g ∈ H => multiply an element from H with any element from G and the result is still in H
∀h ∈ H, ∀g ∈ G : g·h ∈ H => analogue to above


Possible applications I can imagine are:
assume you are programming and you have a finite playing board (like in pacman).
now you add a player, which has a position and who can walk and is connected to the left of the playing field with a spring.
He is very muscular, therefor the spring does not affect his movement at all (and we can model his movement with addition).
Sometimes our player is way too rough though and his spring snaps out of the ground on the left of the field (where our player starts) and catapults him (depending on the strength k of the feather) k times the distance that our player walked away so far.


now he lives in a world where his position lies in a group that is an ideal subgroup of an infinite playing bord, because the force of the spring is nevery able to bring him outside of his world, as it just keeps on wrapping around.
this would be different, if for example the feather would catapult him by i=sqrt(-1), kinda like into another dimension. Then our poor player would not be on his familiar board anymore.
Or, lets assume our player can only move in squares, like in a the classic pokemon red/blue. If the force of the spring would be 1/2 and he stood on a pane/position 3 to the right of the origin, then he would after the spring had snapped be at position half a distance between the first and second pane. But this just is not possible in pokemon and therefor that would not be an ideal!

[xFloppyDolphinn]

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

[franciscovargas]

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

[gggg]

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

[samrichardson]

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!

[jeremyjakob]

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

[iliyakantoma]

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

[lukadeclerck]

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

[bhumikabaddhan]

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

[soulysouly]

"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