I am studying up for the prelims, and I am trying to remember all the things my professors taught me back before summer XD. So! Whilst doing so, I stumbled across the following:
This question concerns the polynomial ring $R = \mathbb{Z}[x, y]$ and the ideal $I = (5, x^2 + 2)$ in $R$.
(a) Prove that $I$ is a prime ideal of $R$ and that $R/I$ is a PID.
(b) Give an explicit example of a maximal ideal of $R$ which contains $I$.
(Give a set of generators for such an ideal.)
(c) Show that there are infinitely many distinct maximal ideals in $R$ which contain $I$.
So, what I think I should do, and what I consider mild sorcery, is to say that
$\mathbb{Z}[x,y]/(5, x^2+2) \cong \mathbb{Z}_5[x,y]/(x^2+2)$.
Now, $x^2 + 2$ has no roots in $\mathbb{Z}_5$ and, as such, is irreducible. Thus, $\mathbb{Z}_5[x]/(x^2 + 2) \cong \mathbb{Z}_5[\sqrt{-2}]$ is a field, and so, we have that $\mathbb{Z}_5[x,y]/(x^2+2) \cong \mathbb{Z}_5[\sqrt{-2}][y]$, which are just polynomials over the field $\mathbb{Z}_5[\sqrt{-2}]$, which is a Euclidean domain, and as such is an Integral Domain, which implies Prime~ness of the Ideal, and is also a PID, proving (a).
So, my question is, are these manipulations valid?!? Haha. If so, can someone explain to me why I can just mod by a single generator of the ideal at a time? I feel it is something that my professor taught me, but I don't see the justification for it.
Additionally, to do parts (b) and (c) I assume you just consider $M = (5, x^2 + 2, f(y))$, where $f(y)$ is an irreducible polynomial in $\mathbb{Z}_5[\sqrt{-2}]$.
Thanks immensely for all your help in advance! =)
The rigorous justification of cancellation in stages is the third isomorphism theorem for commutative rings. Part of it states that for any commutative ring $R$ and any two ideals $I,J$ with $I\subset J$, $$R/J \simeq (R/I)/(J/I).$$ In your case $J = (5,x^2+2)$ and $I=(5)$.
Also, you do not need to invoke Euclidean property at any point. For any field $k$, the polynomial ring in one variable $k[x]$ is an integral domain.