Ring Theory 1: Ring Definition and Examples

Introduction to rings: defining a ring and giving examples. Knowledge of sets, proofs, and mathematical groups are recommended.
Practice problems:
1.) Which of the following sets are rings? If it is, prove it. If not, say which property of rings fails for that set (there may be more than one).
a) N = {1,2,3,4,5,...} under normal addition and multiplication
b) A = {a+b*sqrt(2) | a,b are rationals} under normal addition and multiplication
c) B = {all polynomials p(x) with integer coefficients}
d) C = {all polynomials p(x) with integer coefficients, where deg( p(x) ) is even}
2.) The set {0,2,4,6,8,10,12} is a ring with unity under the operations of addition mod 14 and multiplication mod 14. What is the unity of this ring?
3.) Let R be a commutative ring with unity, and let U(R) denote the set of units (elements with multiplicative inverses) of R. Prove that U(R) is a group under the multiplication defined in R.

[@soniaabraham8728]

why can't professors be this precise and to the point??? Thanks a ton

[@BlackRose4MyDeath]

Because good professors are rare.

[@Suav58]

I had a very good mathematics teacher at high school. He kept in touch with a cutting edge mathematics of the time (70-75'). He sometimes whispered one or other remark, like "Bourbakit tells me to do it like this..." What I appreciated most is opening horizons beyond school mathematics, being in love with the subject and liking us, pupils.

[@codyplayground]

Thank you very much!! You have no idea how many different videos and pages trying to explain the idea of rings and fields and you explained in a way I actually understand. Group theory is not an easy subject to see for many of us.

[@sunaghani7802]

Best videos Ive found in abstract algebra. Thank you so much!! You should do more!!!

[@asifaman72]

Not only (-1), but (1) also possess multiplicative inverse in integers.

[@maxpercer7119]

correct, that was a mistake

[@mcmcshaneo]

Thanks, clearest explanation I found. It helps to just conform the basic things too.

[@vegetaza2384]

This was nicely explained thanks man!

[@sheikhtijanfaye3208]

Proff i really like yours videos. keep it up

[@magedashraf2053]

in 2021 want to tell you its great and usefull ! thanks ;')

[@marieedwards5512]

Great video, was able to understand full definition of rings. Where can I get the solution to the practice questions in the description of the video?

[@Suav58]

In your brain. Don't overegg it. Just read a question, write it down on a clean white sheet of paper. Write down the axioms (again, I hope, you have written them down while listening to the lecture). The answer will come and not by poring over this sheet of paper, but by staying interested, fascinated even. You'll find out, that at some point you will come back to this sheet of paper and write the answer down with ease. The questions aren't very hard.

[@maxpercer7119]

the even integers under addition and multiplication is a ring without unity (or rng)

[@ashutoshnarvariya7833]

Where is the answers of practice problems??

[@Algebrism]

Thank you. It's been changed.

[@alexvandekleut1437]

for the first proof couldnt you stop at a*0 + 0 = a*0then(a+1)*0 =a*0 for any a and thus it is proved by induction?

[@shawnpavlik1293]

Do you have any videos or resources that show that the characteristics of rings are true, i.e. closure under + and *, commutativity under addition and multiplication, additive inverse and identity properties, left and right distributivity, and associativity under +?

[@shawnpavlik1293]

Proofs of those properties, I mean...

[@Lexivor]

These properties constitute the definition of rings and thus don't have proofs. It's like asking for a proof that even numbers are divisible by two, that's true just because of the definition of even.

[@shawnpavlik1293]

Why are you being a douchebag?

[@shawnpavlik1293]

Well, as part of my class, we were asked to prove these 8 properties. I just needed help getting started on one....

[@navjotsingh2251]

Shawn Pavlik there are many abstract algebra books freely available online that all approach proofs a bit differently. It’s hard to say which to recommend because it depends on what level of algebra you are at and also your learning style as books layouts may vary.

[@GlitchE1]

Nice work!

[@Pizner]

your videos are sooooo good

[@RANVEERSINGH-ly5ee]

Good job sir..:)

[@athenstube]

Great work! Thank you.

[@lemonsnpickles]

thanks! very clearly explained :)

[@whomayshebe1]

One thing: at 17:21 you claim that -1 is the only integer that has a multiplicative inverse in Z. Of course 1 also has a multiplicative inverse as well.

[@eduardoxfurtado]

Good job!

[@baseballchew22]

awesome vid, thanks a lot!

[@person1860]

Voice isn't load enough,

[@SuperYtc1]

+person1860 Why isn't this comment louding for me..?

[@richardv6294]

Loud*

[@quantumjuan5192]

I believe you do need a multiplicative identity in a ring. You may be thinking of a multiplicative inverses for each nonzero element in the set.

[@FarnhamJ07]

It depends on who you ask; the field is pretty split on whether a 1 is required or not. He really should've mentioned this! Obviously go with whatever your textbook/prof says for class but definitely be aware that it isn't a settled, solidly defined matter.
Also, fun fact; you'll see some call a ring without multiplicative identity a 'rng', as in 'ring' without the _i_ .

[@paulpjoby444]

awesome :)

[@ousen3629]

Do more niqpgdba

[@sgjbslover]

thank you.
suggestion: speak more openly, like you had no mic and a big audience

[@cmavr8]

Typo alert: "Definiton" in title...

[@maxpercer7119]

how is that a typo?

[@cmavr8]

@@maxpercer7119 it's been corrected!