Factorizations of polynomials with integral non-negative coefficients

We study the structure of the commutative multiplicative monoid ${\mathbb{N}}_0[x{]}^{\ast }$ of all the non-zero polynomials in $\mathbb{Z}[x]$ with non-negative coefficients. We show that ${\mathbb{N}}_0[x{]}^{\ast }$ is not a half-factorial monoid and is not a Krull monoid, but has a structure very similar to that of Krull monoids, replacing valuations into ${\mathbb{N}}_0$ with derivations into ${\mathbb{N}}_0$. We study ideals, chain of ideals, prime ideals and prime elements of ${\mathbb{N}}_0[x{]}^{\ast }$. Our monoid ${\mathbb{N}}_0[x{]}^{\ast }$ is a submonoid of the multiplicative monoid of the ring $\mathbb{Z}[x]$, which is a left module over the Weyl algebra $A_1(\mathbb{Z})$.