These are so fantastically prepared. There are about 27 hours of videos in this playlist. As soon as I'm finished I'll be going through your other series!
07:40 What if I already have the two tables (orange and green), and what I want to find instead is the isomorphism φ that does the proper "relabelling"? Is there any algorithmic method of coming up with such φ? Or at least figuring out that there isn't any?
Sir, i have a doubt. Is the following question even valid. "Consider two non zero rings R and R'. Give an example of a homomorphism f:R-----R' such that R has unity but R' does not have unity?. " Because if R' doesnot have unity then it is not even a ring.
There are several possible definitions of a ring. Some people allow rings not to have unity. A more modern definition of a ring requires that it has unity. If your book/professor is asking for a ring without unity, they are using an older (but still commonly used) definition of a ring. If you want, you can think of this as an algebraic structure which satisfies all of the axioms of a ring except for the axiom requiring the multiplicative identity. A nice situation of a "ring without unity" is an ideal! For any ring R, if I is a nonzero proper ideal of R, then I is a "ring without unity." Can you think of an example of a homomorphism from a ring R to one of its nonzero proper ideals? (Notice that since your codomain has no multiplicative identity, you also drop the condition that a ring homomorphism must send the multiplicative identity to the multiplicative identity.)
These are so fantastically prepared. There are about 27 hours of videos in this playlist. As soon as I'm finished I'll be going through your other series!
Your presentations are absolutely beautiful. Thank you so dearly. Sincerely from Texas
Such a clear and intuitive way of teaching this. Great video.
I am very happy that I found your channel. it helps me a lot for may courses.
07:40 What if I already have the two tables (orange and green), and what I want to find instead is the isomorphism φ that does the proper "relabelling"? Is there any algorithmic method of coming up with such φ? Or at least figuring out that there isn't any?
Sir, i have a doubt. Is the following question even valid.
"Consider two non zero rings R and R'. Give an example of a homomorphism f:R-----R' such that R has unity but R' does not have unity?. " Because if R' doesnot have unity then it is not even a ring.
There are several possible definitions of a ring. Some people allow rings not to have unity. A more modern definition of a ring requires that it has unity. If your book/professor is asking for a ring without unity, they are using an older (but still commonly used) definition of a ring. If you want, you can think of this as an algebraic structure which satisfies all of the axioms of a ring except for the axiom requiring the multiplicative identity.
A nice situation of a "ring without unity" is an ideal! For any ring R, if I is a nonzero proper ideal of R, then I is a "ring without unity." Can you think of an example of a homomorphism from a ring R to one of its nonzero proper ideals? (Notice that since your codomain has no multiplicative identity, you also drop the condition that a ring homomorphism must send the multiplicative identity to the multiplicative identity.)
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