Factorizations of polynomials with integral non-negative coefficients

Abstract

We study the structure of the commutative multiplicative monoid $$\mathbb {N}_0[x]^*$$ of all the non-zero polynomials in $$\mathbb {Z}[x]$$ with non-negative coefficients. The monoid $$\mathbb {N}_0[x]^*$$ is not half-factorial 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 $$\mathbb {N}_0[x]^*$$ . Our monoid $$\mathbb {N}_0[x]^*$$ is a submonoid of the multiplicative monoid of the ring $$\mathbb {Z}[x]$$ , which is a left module over the Weyl algebra $$A_1(\mathbb {Z})$$ .