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Given $R$ is the Ring $\frac{\mathbb{C}[x]}{(x^2+1)}$ .Then which of the following option is correct

$1. $$R$ has exactly two prime ideal

$2.$$R$ is UFD

$3.$$(x)$ is a maximal ideal of $R$

My attempt : $x^2+1 =(x-i)(x+i)$ so option $1$ is true that is R has exactly two prime ideal

$\frac{\mathbb{C}[x]}{(x^2+1)}$is isomorphics to $\mathbb{C}^2$ so we know that $\mathbb{C^2} $is euclidean domain so it will be UFD

option 3 is obviously false

Is its correct ??

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    $\begingroup$ It is $\mathbb{C} \times \mathbb{C}$, which is not even an integral domain. But it has exactly two prime ideals. $\endgroup$ – Sunny Jun 10 '19 at 11:08
  • $\begingroup$ Hi: i've edited your title to be more useful. Please consider carefully your future titles and make sure they are informative, rather than excuses for titles. $\endgroup$ – rschwieb Jun 10 '19 at 13:50
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1) Your work in this is OK, but you haven't really justified why there are two prime ideals. Merely factoring into two rings does not guarantee that. It would be better to clarify how you arrive at that from your isomorphism with $\mathbb C\times \mathbb C$.

2) No, it is not even a domain.

3) If it is obvious then it should cost you little to give a simple reason. What is your reason? For an exercise in class, few people are going to have patience with someone who answers "obvious."

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