On the dimension of polynomial semirings

Abstract

In our previous work, motivated by the study of tropical polynomials, a definition for prime congruences was given for an arbitrary commutative semiring. It was shown that for additively idempotent semirings this class exhibits some analogous properties to prime ideals in ring theory. The current paper focuses on the resulting notion of Krull dimension, which is defined as the length of the longest chain of prime congruences. Our main result states that for any additively idempotent semiring A, the semiring of polynomials dim⁡A[x1,…,xn] and the semiring of Laurent polynomials A[x1±1,…,xn±1], we have dim⁡A[x1±1,…,xn±1]=dim⁡A[x1,…,xn]=dim⁡A+n.