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I am self-studying commutative algebra, and came across the following scenario but am rather stuck. Any hints/advice would be greatly appreciated.

Consider R[X], the polynomial ring over commutative ring R in indeterminate X, where R is not necessarily a field. Let M be a maximal ideal of R.
1) Is there any correlation between the maximal ideals of R[X] and those of R?
2) Which maximal ideal of R[X] contains M[X]?

So from the second part of this question, I can infer that there is not a trivial link between maximal ideals of each structure (since M[X] is not a maximal ideal of R[X]), but I cannot see an explicit connection. I am familiar with the Nullstellensatz but that result was only applicable to fields (and is dependent on that fact) so perhaps I need to explore an entirely different avenue? Thank you in advance.

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  • $\begingroup$ 1) I don't think so. 2) Look for maximal ideals of $R[X]/M[X]\simeq (R/M)[X]$. $\endgroup$ – user26857 Sep 10 '16 at 18:10
  • $\begingroup$ For the first question consider the case when $R$ is a (not algebraically closed) field. Are you thinking in a similar "correlation"? Even in this case you'll see that it is not much. $\endgroup$ – Heinrich Sep 10 '16 at 18:42
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Let $M$ be a maximal ideal of $R$, and $N$ a maximal ideal of $R[X]$ which contains $M$, since $N$ is an ideal, $M[X]\subset N$. There exists a map $p_M:R[X]\rightarrow (R/M)[X]$ such that $p_M(X)=X$ and the restriction of $p_M$ to $R$ is the quotient map $R\rightarrow R/M$. Since $R/M$ is a field, the maximal ideals of $(R/M)[X]$ are defined by irreducible polynomials of $(R/M)[X]$. If $N$ is a maximal ideal of $R[X]$ which contains $M$, $p_M^{-1}(p_M(N))=N$. We deduce that the maximal ideal which contains $M$ corresponds to monic irreducible polynomials of $(R/M)[X]$.

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  • $\begingroup$ Thank you. I am just a bit uncertain why R/M is necessarily a field? EDIT: I just realized it follows from the fact that M is a maximal ideal. $\endgroup$ – Jan Alwyn Sep 12 '16 at 11:58

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