On the dimension of polynomial semirings

TL;DR

This paper investigates the Krull dimension of polynomial and Laurent polynomial semirings over additively idempotent semirings, establishing that their dimensions are exactly one greater than that of the base semiring.

Contribution

It proves that for additively idempotent semirings, the Krull dimension of polynomial and Laurent polynomial semirings equals the base dimension plus one, extending the understanding of prime congruences.

Findings

Dimension of A[x] equals dimension of A plus one.

Dimension of A(x) equals dimension of A plus one.

Prime congruences in additively idempotent semirings behave analogously to prime ideals in rings.

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 $A[x]$ and the semiring of Laurent polynomials $A(x)$, we have $\dim A[x] = \dim A(x) = \dim A + 1$.