Clarification for the prime ideal $\langle x\rangle$

Question

$$\frac{\mathbb Z[x]}{\langle x\rangle}=\{g(x)+\langle x\rangle\,|\,g(x)\in Z\,[x]\}$$

But $\langle x\rangle$ absorbs all non-constant polynomials thus:

$$\frac{\mathbb Z[x]}{\langle x\rangle}=\{a+\langle x\rangle\,|\,a\in Z\}$$

This conclusion can be made correct?

Answer 1

That is indeed a way to think about it, and your description of $\Bbb Z[x]/\langle x \rangle$ is correct.