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.
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 $(R/M)[X]$.