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

Abstract

The study of monic and monic free ideals in a polynomial semiring S [ x ] S[x] , where S S is a commutative semiring with an identity, is extended to S [ x 1 , x 2 , … , x n ] S[{x_1},{x_2}, \ldots ,{x_n}] . Structure theorems are given for monic, monic free, and monic free k k -ideals in S [ x 1 , x 2 , … , x n ] S[{x_1},{x_2}, \ldots ,{x_n}] . It is shown that each monic free k k -ideal in S [ x 1 , … , x n ] , S S[{x_1}, \ldots ,{x_n}],\;S a strict semiring, is the sum of a finite number of ideals B i {B_i} such that each B i {B_i} is the union of a proper infinite ascending chain of ideals.