What Are the Ring Homomorphisms from Z[X] to Z[X]?

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Discussion Overview

The discussion revolves around the nature of ring homomorphisms from the polynomial ring Z[X] to itself, as well as to other rings. The scope includes theoretical aspects of ring homomorphisms and their properties.

Discussion Character

  • Exploratory, Debate/contested, Homework-related

Main Points Raised

  • One participant asks for all ring homomorphisms from Z[X] to Z[X], indicating a desire for a comprehensive understanding of the topic.
  • Another participant suggests that understanding homomorphisms from Z[X] to any ring R is a prerequisite for addressing the original question, implying a broader context is necessary.
  • A participant expresses frustration at what they perceive as a redundant remark, emphasizing their genuine request for help rather than conjecture.
  • In response, another participant clarifies that their intention was to highlight a powerful theorem related to homomorphisms from polynomial rings, suggesting that a general approach may be more beneficial.
  • One participant acknowledges the feedback and indicates they will review the relevant material.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the best approach to the problem, with differing views on how to frame the discussion and the relevance of broader contexts.

Contextual Notes

There are indications of missing assumptions regarding the definitions and properties of ring homomorphisms, as well as unresolved mathematical steps related to the general case of homomorphisms.

esisk
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what are all the ring homomorphism from Z[X] to Z[X]. Thank you
 
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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.
 
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
 
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?)
 
Okay, thank you. I will review
 

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