Factorizations of polynomials with integral non-negative coefficients

We study the structure of the commutative multiplicative monoid N 0 [ 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 ) .