On the Content of Polynomials Over Semirings and Its Applications

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.

arXiv commentFinal version published at J. Algebra Appl., one reference added, three minor editorial changes

Cited by in corpus (10)

• Valuation Semirings
• Some remarks on semirings and their ideals
• On 2-absorbing ideals of commutative semirings
• On zero-divisors of semimodules and semialgebras
• Invertible Ideals and Gaussian Semirings
• Algebraic properties of expectation semirings
• Distinguished elements in semiring extensions
• Eversible and reversible semigroups and semirings
• Classical semirings
• Algebraic properties of Indigenous semirings