In my D&F homework I proved that the ring $R = \mathbb{Z} + x\mathbb{Q}[x]$, $\mathbb{Q}$-polynomials with integral constant terms is an integral domain. I cannot help but feel that $R$ is a very “strange” ring. Is it possible to express $R$ as a $R'/P$, where $R'$ is a more familiar ring and $P \subset R'$ is a prime ideal?
We also considered the ring $R/(x)$ and expressed it as $\left\{a_0 + a_1x \,\middle|\,a_0 \in \mathbb{Z}, a_1\in \mathbb{Q}/\mathbb{Z}\right\}$, with $\mathbb{Q}/\mathbb{Z}$ considered as the additive group of rational numbers up to integer difference. Cross-terms arising in multiplication in $R/(x)$ are handled using the obvious action $\mathbb{Z} \times \mathbb{Q} \to \mathbb{Q}$, and quadratic terms vanish.
I am just a little bit uncomfortable using a group in the coefficients of polynomials in a ring. Is there a way to construct $R/(x)$ using more mundane operations like direct sums, tensor and semidirect products, and/or group rings?
Finally, are there more general properties I should know about polynomial rings where higher terms take coefficients from the localization of the coefficient ring of the lower terms?
Consider the ring homomorphism $\phi: \mathbb Q[x] \to \mathbb Q$ given by $\phi(f)=f(0)$.
Then $R = \phi^{-1}(\mathbb Z)$.
In general, given a ring homomorphism $\phi: A \to B$ and a subring $B' \subseteq B$, then $\phi^{-1}(B')$ is a subring of $A$.