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I have question which I am not sure wether or not it is correct.

find all the rings which is isomorphic to $Z[x]$ as ring. I believe, we can't find them without giving specific conditions about the ring.

Any help will be appreciated

Team
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  • Up to isomorphism, there is only one such ring. This is a weird question, $Z[y]$ is isomorphic to $Z[x]$ if you want. I can use a lot more letters and produce many isomorphic rings. – Mathematician 42 Feb 01 '17 at 13:31

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It makes no sense to ask for "all the rings isomorphic to $\mathbb{Z}[X]$". There are as many as there are ways to give the elements of that ring different names. It's like asking for "all the six element sets".

It does make sense to ask if any particular ring is isomorphic to that one, or whether any particular list of rings has any that look like $\mathbb{Z}[X]$. That's pretty much what your second sentence hints at.

Loosely speaking, "is isomorphic to" is the same as "looks just like". See What is an Homomorphism/Isomorphism "Saying"? for more on this point of view.

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Ethan Bolker
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  • Thanks, do you mean there is no way find them? Is it right – Team Feb 01 '17 at 13:51
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    Yes, there is no way to find them. It's even worse: there's really no sensible way to ask the question. Think about "can you find all the six element sets?" – Ethan Bolker Feb 01 '17 at 14:09
  • @Team You're welcome. If the answer satisfies you, you can accept it (check mark) and (if you wish) upvote it (up arrow). – Ethan Bolker Feb 01 '17 at 15:28