Ring Homomorphism: Z[X] to Z[X]

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In summary, the conversation is about finding all the ring homomorphisms from Z[X] to R for a ring R. The person asking for help is reminded of a theorem that can be helpful in solving the problem, and is thanked for the reminder.
  • #1
esisk
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what are all the ring homomorphism from Z[X] to Z[X]. Thank you
 
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  • #2
What are all the ring homomorphisms from Z[X] to R for a ring R? If you can't do that, then you can't do this problem.
 
  • #3
Dear Hurkyl,
While I (I hope) do see the point behind you remark, quite frankly I do not appreciate it as it is redundant. I am genuinely asking for help and you are conjecturing when one is not able to solve this problem. I thank you
 
  • #4
It looked like a request for an answer, not a request for help -- it is strictly against our policy to hand out answers to homework problems.

My response was meant in all seriousness -- you should have seen in your class a very powerful theorem about homomorphisms from polynomial rings. And sometimes, it's easier to find/work out a solution when you consider the general case without regard to the specific. I assume from your response that you were able to recall the answer? (or able to track it down in your book?)
 
  • #5
Okay, thank you. I will review
 

1. What is a ring homomorphism?

A ring homomorphism is a function between two rings that preserves the ring structure. This means that it maps the operations of addition and multiplication in one ring to the corresponding operations in the other ring.

2. Can you give an example of a ring homomorphism from Z[X] to Z[X]?

Yes, the function f: Z[X] -> Z[X] defined by f(a + bx) = a - bx is a ring homomorphism. It maps the addition of polynomials in Z[X] to the subtraction of polynomials in Z[X], and the multiplication of polynomials in Z[X] to the multiplication of polynomials in Z[X].

3. What is the kernel of a ring homomorphism?

The kernel of a ring homomorphism is the set of elements in the domain that are mapped to the additive identity in the codomain. In other words, it is the set of elements that are mapped to zero by the homomorphism.

4. Is the kernel of a ring homomorphism always an ideal?

Yes, the kernel of a ring homomorphism is always an ideal in the domain. This is because it is a subset of the domain that is closed under addition and multiplication, and it contains the additive identity element.

5. Can a ring homomorphism be surjective but not injective?

Yes, a ring homomorphism can be surjective (onto) but not injective (one-to-one). This means that the homomorphism maps every element in the codomain to, but there may be multiple elements in the domain that are mapped to the same element in the codomain.

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