Here's a proposal for a proof, though I can't finish it.
- A morphism in a category is a regular epimorphism if it coequalizes some pair of morphisms.
- An object $P$ in a category is projective if $\mathrm{Hom}(P,X)\to \mathrm{Hom}(P,Y)$ is surjective for every regular epimorphism $X\to Y$.
- An object $K$ in a category is compact if $\mathrm{Hom}(K,-)$ takes filtered colimits to filtered colimits of sets.
- An object $X$ in a category is irreducible if the only retracts of $X$ are itself and the initial object.
A self-equivalence of a category will preserve the these conditions.
Claim: $\mathbb{Z}[x]$ is (up to isomorphism) the unique irreducible compact projective in commutative rings.
What I do know (I think):
- Regular epis in $\mathrm{Rings}$ are exactly the surjective ring homomorphisms.
- Compact objects in $\mathrm{Rings}$ are exactly the finitely presented rings.
Since polynomial rings are certainly projective, this means that the compact projective objects in $\mathrm{Rings}$ are precisely the retracts of the $\mathbb{Z}[x_1,\dots,x_n]$s.
My intuition is that retracts of a polynomial ring are always isomorphic to a polynomial ring. In which case $\mathbb{Z}[x]$ would be the only irreducible compact projective. But I could be completely wrong about that.