A question about ring homomorphisms

In summary, a ring homomorphism is a function between two rings that preserves the ring structure. It differs from a group homomorphism in that it also preserves multiplication. The kernel of a ring homomorphism is the set of elements in the first ring that are mapped to the identity element in the second ring. A ring homomorphism can be either surjective or injective. Real-world applications of ring homomorphisms include cryptography, quantum mechanics, and computer science.
  • #1
Artusartos
247
0
I attached a page from my textbook, because there was something that I didn't understand.

What I don't understand is in the proof it says let f(x) be...etc. but in the theorem, it says nothing about f(x). In other words, where in the thoerem does it say anything about f(x). Why are they talking about f(x) in the proof?

Thanks in advance
 

Attachments

  • 20121224_221945.jpg
    20121224_221945.jpg
    29.4 KB · Views: 398
  • 20121224_222018.jpg
    20121224_222018.jpg
    34.7 KB · Views: 384
Physics news on Phys.org
  • #2
Notice that f(x) is just a polynomial in R[x] and the author(s) use it to define the map claimed in the proposition.
 

1. What is a ring homomorphism?

A ring homomorphism is a function between two rings that preserves the ring structure. This means that for any two elements in the first ring, the function will produce the same result as if the two elements were operated on directly in the second ring.

2. How is a ring homomorphism different from a group homomorphism?

While both ring homomorphisms and group homomorphisms are functions that preserve structure, the main difference is that ring homomorphisms also preserve multiplication in addition to addition and subtraction. In other words, a ring homomorphism is a group homomorphism with an additional requirement of preserving multiplication.

3. What is the kernel of a ring homomorphism?

The kernel of a ring homomorphism is the set of elements in the first ring that are mapped to the identity element in the second ring. In other words, it is the set of elements that are mapped to 0 under the ring homomorphism.

4. Can a ring homomorphism be surjective?

Yes, a ring homomorphism can be surjective, meaning that every element in the second ring is mapped to by at least one element in the first ring. However, it is not always the case and a ring homomorphism can also be injective, where different elements in the first ring may map to the same element in the second ring.

5. Are there any real-world applications of ring homomorphisms?

Yes, ring homomorphisms have many applications in mathematics and other fields. For example, they are used in cryptography to encrypt and decrypt messages, in quantum mechanics to model symmetries, and in computer science for data compression and error correction.

Similar threads

I Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
    • Calculus
Replies
9
Views
849
I Substitution in a definite integral
    • Calculus
Replies
6
Views
1K
B Question about the fundamental theorem of calculus
    • Calculus
Replies
2
Views
294
I Some questions on l'Hôpital rules
    • Calculus
Replies
9
Views
2K
B Why is Big-O about how rapidly the Taylor graph approaches that of f(x)?
    • Calculus
Replies
19
Views
2K
B Functions which relate to calculus: Questions about Notation
    • Calculus
2
Replies
36
Views
4K
B Why do I not understand what seems to be basic Calculus?
    • Calculus
Replies
11
Views
2K
MHB Intermediate Value Theorem
    • Calculus
Replies
5
Views
957
I Understanding Theorem 13 from Calculus 7th ed, R. Adams, C. Essex, 4.10
    • Calculus
Replies
10
Views
2K
I Open interval or Closed interval in defining convex function
    • Calculus
Replies
4
Views
2K

Hot Threads

Recent Insights

Back
Top