Polynomial ring modulo ideal generated by linear factor

Question

How do you prove that, for any integer $n$,
$$ℤ[X]/(X-n) ≅ ℤ$$

The following straightforward way seems arduous: write an arbitrary $∑_i a_iX^i ∈ ℤ[X] $ as an integer plus $X - n$. I wouldn't know how to do that.

Smarter is probably to use the First Isomorphism Theorem: find a homomorphism $f : ℤ[X] \rightarrow ℤ[X]$ with $\ker f = (X -n)$ and im$f = ℤ$. Is that smarter, though, or is that just a rephrasing of the first problem?

Could this be done in the same manner for all commutative rings, and not just ℤ?

Answer 1

It is valid for every commutative ring $R$ and linear polynomial $X-r$, and the first isomorphism theorem is a good approach.

But you should use it for a homomorphism $R[X] \to R$.

$\sum_i a_iX^i\mapsto \sum_i a_ir^i$