= Solution
Noetherian means every <ideal> is finitely generated, equivalently every ascending <ideal> chain stabilises. Quotients preserve this; for example $\mathbb Z[\sqrt{-5}]$ is Noetherian as a quotient of $\mathbb Z[X]$ but not a UFD. A UFD need not be Noetherian (a <polynomial> <ring> in infinitely many variables is a counterexample). Stabilisation of kernels proves every surjective endomorphism of a <Noetherian ring> injective. The shift $k[x_1,x_2,\ldots]\to k[x_1,x_2,\ldots]$, $x_1\mapsto0$, $x_{i+1}\mapsto x_i$, is surjective noninjective; $x\mapsto x^2$ on $k[x]$ shows injective need not mean surjective. Finally $f_n(X)=\binom Xn$ is integer-valued. Finite differences prove uniquely that every integer-valued <polynomial> is an <integral> linear combination of the $f_n$, so they are a $\mathbb Z$-basis. Int$(\mathbb Z)$ is not Noetherian: the denominators in $f_p$ yield an <ideal> chain requiring new prime denominators.
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