Let $A$ be a commutative ring, $\mathbb{q}\subset A$ an ideal of $A$, and $\mathbb{q}A[x]$ the ideal of $A[x]$ generated by $\mathbb{q}$ (consists of the polynomials with coefficients in $\mathbb{q}$). Show $\mathbb{q}$ is prime in $A \Leftrightarrow \mathbb{q}A[x]$ is prime in $A[x]$.
I've attempted to construct an isomorphism from $(A/\mathbb{q})[x]\to A[x]/\mathbb{q}A[x]$, but I'm not really sure how to go about doing so other than some appeal to the first isomorphism theorem for rings.
Just consider the natural map $A[x]\rightarrow A/q[x]$ and note that its kernel is exactly $qA[x]$.
To prove that $qA[x]$ is again prime you have to prove that for $P\cdot Q \in qA[x]$ either $P$ or $Q$ is in $qA[x]$. Induction on the sum of the degrees of $n=deg(P)$ and $m=deg(Q)$. If $n=0$ it follows from the fact that $q$ is prime. If $n>0$ $$ PQ = a_{n}b_{m}X^{n+m}+\ldots $$ so $a_{n}b_{m}\in q$ and wlog $a_{n}\in q$. Now $P=P' +a_{n}X^{n}$ and $$ PQ = P'Q+a_{n}X^nQ\in qA[x]\Rightarrow P'Q\in qA[x] $$ Now use the induction hypothesis!
We can assume $\mathfrak q=0$ (replace $A$ by $A/\mathfrak q$). Then the statement reduces to:
$A$ is a domain if and only if $A[x]$ is a domain,
which is clear.
Hint: $q$ prime $\:\Rightarrow\:qA[x]$ prime follows from examining leading coefficients mod $q$, i.e.
${\rm mod}\ q:\: f,g \not\equiv 0\:\Rightarrow f \equiv a\:x^n+\cdots,\ \ g \equiv b\:x^m+\:\cdots,\ a,b\not\in q,\:$ where $\cdots$ means lower degree terms. Hence we deduce $\:fg \equiv ab\:x^{n+m}+\cdots,\ ab\not\in q\:\Rightarrow\: fg \not\equiv 0$.
Note: for $q = 0$ the proof becomes: domain $A\:\Rightarrow\:$ domain $A[x]$. The proof works by exploiting the multiplicativity of the leading coefficient map combined with the nonexistence of zero-divisors in a domain, to show that the product of nonzero polynomials remains nonzero because the same holds true for their leading coefficients.
This is a prototypical example of using a multiplicative map to transfer multiplicative properties from one monoid to another (here the nonexistence of zero-divisors). For some further examples, see this post on using norms to transfer multiplicative properties between $\mathbb Z$ and rings of algebraic integers, and also this post where one deduces the existence of inverses of domain elements $\ne 0$ that are algebraic over a subfield (generalizing rationalizing denominators).
Note: one can reduce to the case $A$ is a domain by factoring out by the prime $q$. However, when learning these techniques, it is instructive to compare the structural vs. element-wise proofs.
Note also that this may be viewed as a generalization of one form of Gauss's Lemma.
