On the Content of Polynomials Over Semirings and Its Applications

TL;DR

This paper explores the algebraic structure of polynomials over semirings, introducing new classes like Gaussian and weak Gaussian semirings, and investigates their properties, including prime ideals and zero-divisors, with applications to content semialgebras.

Contribution

It introduces Gaussian and weak Gaussian semirings, generalizes content semialgebras, and analyzes prime ideals and zero-divisors in polynomial and power series semirings over semirings.

Findings

Dedekind-Mertens lemma holds only for subtractive subsemimodules.

Bounded distributive lattices are Gaussian semirings.

Formal power series semirings can be weak content semialgebras.

Abstract

In this paper, we prove that Dedekind-Mertens lemma holds only for those semimodules whose subsemimodules are subtractive. We introduce Gaussian semirings and prove that bounded distributive lattices are Gaussian semirings. Then we introduce weak Gaussian semirings and prove that a semiring is weak Gaussian if and only if each prime ideal of this semiring is subtractive. We also define content semialgebras as a generalization of polynomial semirings and content algebras and show that in content extensions for semirings, minimal primes extend to minimal primes and discuss zero-divisors of a content semialgebra over a semiring who has Property (A) or whose set of zero-divisors is a finite union of prime ideals. We also discuss formal power series semirings and show that under suitable conditions, they are good examples of weak content semialgebras.