Your welcome, and thanks for the kind words! The domain property requires 1) commutativity and 2) no zero divisors. 2) consider Z/6. It has ideals (0), (2), (3), and Z/6, so all are principal. But 2 and 3 are zero divisors since 2 x 3 = 6 = 0. So not a domain. 1) M = nxn matrices over R, C, or any division ring, n>1. Not commutative, but all left and right ideals are principal. Here every ideal is of the form eM or Me where e is an idempotent (e^2=e).
Do you have a video that shows UF "restored" with ideals? For example that shows that every ideal in Z[sqrt(-5)] is the product of a unique set of prime ideals, in particular, for the ideal (6), which unique prime ideal factorization corresponds?
@@MathDoctorBob I just found a video illustrating that very example I was asking. This video happened to be posted just three months ago and we can thank the quarantine for it. ua-cam.com/video/wSE9RcHZDuE/v-deo.html The whole channel have very nice lectures on Algebraic Number Theory Enjoy!
Thanks for the great videos. For the Z[sqrt(-5)] example, aren't those factors irreducible but not prime? It seems like you say "we would still need to show that they're primes."
Good catch! I was thinking "irreducible" which isn't defined until the next board. Of course, this example shows irreducible and prime are not the same. Annotated. Edit: Reading around, I've noticed in the literature some use of prime when irreducible is called for, even in this exact example. YMMV.
hi dr. , how r u ? I do really enjoy ur lectures , I am studying ring theory this semester , I need an example of a principal ideal ring that is not a principal ideal domain, thank u .
+theleastcreative Fraleigh's A First Course in Abstract Algebra is probably the most popular for beginners (not hardcore mode). I grew up on Herstein's Topics in Algbera and Artin's Algebra, which can be tough. People will also mention Aluffi's book and Dummit and Foote, but I haven't spent time with either.
Your welcome, and thanks for the kind words! The domain property requires 1) commutativity and 2) no zero divisors.
2) consider Z/6. It has ideals (0), (2), (3), and Z/6, so all are principal. But 2 and 3 are zero divisors since 2 x 3 = 6 = 0. So not a domain.
1) M = nxn matrices over R, C, or any division ring, n>1. Not commutative, but all left and right ideals are principal. Here every ideal is of the form eM or Me where e is an idempotent (e^2=e).
Your Lectures are excellent!
A field always “has a one” right? 4:41
thanks for the video! really helpful :)
Your welcome! Good luck!
Do you have a video that shows UF "restored" with ideals? For example that shows that every ideal in Z[sqrt(-5)] is the product of a unique set of prime ideals, in particular, for the ideal (6), which unique prime ideal factorization corresponds?
That subject belongs to proper algebraic number theory. I'd love to see a presentation of it with a large number examples broken down.
@@MathDoctorBob thanks for answering. Have a great day
@@MathDoctorBob I just found a video illustrating that very example I was asking. This video happened to be posted just three months ago and we can thank the quarantine for it. ua-cam.com/video/wSE9RcHZDuE/v-deo.html
The whole channel have very nice lectures on Algebraic Number Theory
Enjoy!
@@zy9662 Thanks for sharing! Looks good.
Thanks for the great videos.
For the Z[sqrt(-5)] example, aren't those factors irreducible but not prime? It seems like you say "we would still need to show that they're primes."
Good catch! I was thinking "irreducible" which isn't defined until the next board. Of course, this example shows irreducible and prime are not the same. Annotated.
Edit: Reading around, I've noticed in the literature some use of prime when irreducible is called for, even in this exact example. YMMV.
MathDoctorBob Thanks. I noticed that too. It's confusing!
hi dr. , how r u ? I do really enjoy ur lectures , I am studying ring theory this semester , I need an example of a principal ideal ring that is not a principal ideal domain, thank u .
Is there a 1st textbook on algebra you would recommend to a senior undergrad student?
+theleastcreative Fraleigh's A First Course in Abstract Algebra is probably the most popular for beginners (not hardcore mode). I grew up on Herstein's Topics in Algbera and Artin's Algebra, which can be tough. People will also mention Aluffi's book and Dummit and Foote, but I haven't spent time with either.
MathDoctorBob Aluffi‘s Algebra Chapter 0 is a good book but I don‘t think it‘s suitable as first book on algebra for undergrads
شكرآ لك.
Your Lectures are excellent!