A surjective homomorphism which is not an isomorphism for rings (Using basic algebra)

Let $R=\mathbb{Z}[x_1,x_2,\ldots,x_n,\ldots]$. Consider the map induced by sending $x_1$ to $0\in R$, and sending $x_{k+1}$ to $x_k$ for $k=2,3,4,\ldots$. Verify this gives a surjective ring homomorphism, but that it is not injective (hence not an isomorphism).

Analogous to @Arturo's answer, consider a "left-shift" on the infinite product of $\Bbb Z_n$'s say. That's $(x_1,x_2,\dots)\mapsto(x_2,x_3,\dots)$.

As a corollary, invoking the theorem you mentioned, we have that $\prod_{k=1}^\infty\Bbb Z_n$ is not Noetherian.