Ring Theory 5: Zero Divisors and Integral Domains

Intuition behind zero divisors, and rings without zero divisors.
Practice problems:
1) Find all units of Z_20, and find all zero divisors of Z_20. Do you see a relationship between them?
2) Show that every nonzero element of Z_n is either a unit or a zero divisor.
3) Let d be a fixed integer. Prove that the set {a + b*sqrt(d) | a,b are integers} is an integral domain (under normal addition and multiplication).
4) Cancellation property: Let a,b,c be elements of an integral domain such that a is not zero. Prove that if ab=ac, then b=c. (Note: we didn't assume a is a unit)

[@tokkia1384]

If Sauron had paid attention in his abstract algebra class, he’d have made his ring without zero divisors, and it would never have been destroyed

[@joshuaabraham7308]

I'm stealing this joke

[@mthokozisisteventavinnkosi1994]

9 years later. GREAT VIDEO!

[@afiqahnafz]

I really love the way or the technique of your teaching. It has really been helpful for me. Keep making more video :)

[@lucasm4299]

I actually understand what you’re saying. It makes sense. Zn is an integral domain if n is prime.

[@xpanuzx]

Really helpful video man , thank you very much , keep doing this !!!!

[@mjjeon2292]

Holy Guacamoly, this is the most ELI5 answer I've come across. Thank you so much and may the higher being bless you.

[@igormorgado]

Congrats! You have Salman Khan grip for teaching. You should try submit your videos to them since they lack abstract algebra videos. And by the way. Keep it coming.

[@nyceric95]

Dude from the bottom of my heart, good looks.

[@saranyadevi7423]

the way of teachin is really fantastic.this video really help me to understand the concept better. keep on making these kind of wonderful videos..

[@musicislife665]

finally a great video on polynomial rings. bravo

[@tuirnval]

Nice. A lot of fun in math is around clever uses. I think that is what separated Fibonacci from the academics later on. (Neumann, Hilbert, etc)

[@ahexcuseme6936]

thanks for this video!

[@Obito313]

@1:10 I think it would be a good time to introduce a little number theory, but great video series keep them coming!

[@Miixell1]

Great vid! However, isn't there a generalized formula for finding the amount of zero divisors in a ring??

[@cpyke2008]

thanks!

[@Taricus]

What ever happened to the videos on polynomial rings? @_@

[@DarkdwarfSoaD]

Why is Z7 an integral domain, when 2*7 = 0 in that ring? Clearly there exist zero divisors in the ring, no?

[@Algebrism]

+DarkdwarfSoaD 7=0 in Z_7

[@jamiebear6476]

The elements of Z7 are 0,1,2,3,4,5,6.....

[@igormorgado]

Congrats! You have Salman Khan grip for teaching. You should try submit your videos to them since they lack abstract algebra videos. And by the way. Keep it coming.