Monic and monic free ideals in a polynomial semiring in several variables

The study of monic and monic free ideals in a polynomial semiring $S[x]$, where $S$ is a commutative semiring with an identity, is extended to $S[{x_1},{x_2}, \ldots ,{x_n}]$. Structure theorems are given for monic, monic free, and monic free $k$-ideals in $S[{x_1},{x_2}, \ldots ,{x_n}]$. It is shown that each monic free $k$-ideal in $S[{x_1}, \ldots ,{x_n}],\;S$ a strict semiring, is the sum of a finite number of ideals ${B_i}$ such that each ${B_i}$ is the union of a proper infinite ascending chain of ideals.