LAUSR

We study the structure of the commutative multiplicative monoid $$\mathbb {N}_0[x]^*$$N0[x]∗ of all the non-zero polynomials in $$\mathbb {Z}[x]$$Z[x] with non-negative coefficients. The monoid $$\mathbb {N}_0[x]^*$$N0[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 $$\mathbb {N}_0$$N0 with derivations into $$\mathbb {N}_0$$N0. We study ideals, chain of ideals, prime ideals and prime elements of $$\mathbb {N}_0[x]^*$$N0[x]∗. Our monoid $$\mathbb {N}_0[x]^*$$N0[x]∗ is a submonoid of the multiplicative monoid of the ring $$\mathbb {Z}[x]$$Z[x], which is a left module over the Weyl algebra $$A_1(\mathbb {Z})$$A1(Z).