Universal property of polynomial ring in $\mathbf{CRING}$

Question

I know that the polynomial ring $A[x]$ is the free $A$-algebra on $\{x\}$; this is its universal property in the category of $A$-algebras. Is there also a universal property for $A[x]$ considered as a ring, which uniquely determines $A[x]$ in $\mathbf{CRING}$? Would be very useful, as I grew up in $\mathbf{CRING}$!

Answer 1

The polynomial ring $A [x]$ is the coproduct of $A$ and $\mathbb{Z} [x]$.

One way to see this is to note that the category of (commutative unital) $A$-algebras is isomorphic to the slice category $^{A /} \mathbf{CRing}$, and the forgetful functor $^{A /} \mathbf{CRing} \to \mathbf{CRing}$ has a left adjoint, namely $B \mapsto A \otimes_{\mathbb{Z}} B$. Since adjoints compose, we deduce that the free $A$-algebra on one generator is the coproduct of $A$ and the free commutative ring on one generator.