Factorizations of polynomials with integral non-negative coefficients

We study the structure of the commutative multiplicative monoid N0[x]*of all the non-zero polynomials in Z[x] with non-negative coefficients. The monoid 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 N-0 with derivations into N-0. We study ideals, chain of ideals, prime ideals and prime elements of N-0[x]*. Our monoid N-0[x]* is a submonoid of the multiplicative monoid of the ring Z[x], which is a left module over the Weyl algebra A(1)(Z).