Question: Are the integers $\mathbb{Z}$ an affine $K$-algebra, i.e. does there exist a field $K$, a $n\!\in\!\mathbb{N}$, and an ideal $I\!\unlhd\!K[x_1,\ldots,x_n]\!=\!K[\mathbb{x}]$, such that $\mathbb{Z}\!\cong\!K[\mathbb{x}]/I$, as rings?
If we assume that $I\!=\!0$, then since $\mathbb{Z}$ has two units and $K[\mathbb{x}]$ has $|K|\!-\!1$ units, we must have $K\!=\!\mathbb{Z}_3$. Furthermore, since $\mathbb{Z}$ is a PID, we must have $n\!=\!1$. But $\mathbb{Z}\!\ncong\!\mathbb{Z}_3[x]$, since as an abelian group, $\mathbb{Z}$ is generated by $1$ element, whilst $\mathbb{Z}_3[x]$ is not.
If $I\!\neq\!0$, then $I$ must be prime but not maximal, since $\mathbb{Z}$ is a domain but not a field. Since $\mathbb{Z}$ is not local, $\sqrt{I}$ must not be maximal. This is where I run out of ideas...
Additional question: A group presentation is the free group modulo a normal subgroup, and it is denoted $\langle x_1,\ldots,x_n | w_1,\ldots,w_m\rangle$. An $R$-algebra presentation is the free algebra modulo an ideal, and it is denoted $R\langle x_1,\ldots,x_n|p_1,\ldots,p_m\rangle$. Is the commutative $R$-algebra presentation $R[x_1,\ldots,x_n]/I$, where $I$ is the ideal generated by $p_1,\ldots,p_m$, by any chance denoted by $R[x_1,\ldots,x_n|p_1,\ldots,p_m]$? I have not seen this anywhere in the literature. Is this notation reserved for something else?