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Distinguished elements in semiring extensions

Peyman Nasehpour

Abstract

In this paper, we investigate zero-divisor, nilpotent, idempotent, unit, small, and irreducible elements in semiring extensions such as amount, content, and monoid semialgebras. We also introduce new concepts such as the prime avoidance property in semirings, entire-like semirings, semialgebras with Property (A), and also, Armendariz and McCoy semialgebras and we prove some results related to these concepts. For example, we prove that if $B$ is an $S$-semialgebra, then under some conditions, the set of zero-divisors $Z(B)$ of $B$ is the union of the extended maximal primes of $Z(S)$. Finally, we prove a generalization of Eisenstein's irreducibility criterion.

Distinguished elements in semiring extensions

Abstract

In this paper, we investigate zero-divisor, nilpotent, idempotent, unit, small, and irreducible elements in semiring extensions such as amount, content, and monoid semialgebras. We also introduce new concepts such as the prime avoidance property in semirings, entire-like semirings, semialgebras with Property (A), and also, Armendariz and McCoy semialgebras and we prove some results related to these concepts. For example, we prove that if is an -semialgebra, then under some conditions, the set of zero-divisors of is the union of the extended maximal primes of . Finally, we prove a generalization of Eisenstein's irreducibility criterion.

Paper Structure

This paper contains 7 sections, 44 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Amount semialgebras
  3. Semialgebras having Property (A)
  4. Armendariz and McCoy semialgebras
  5. Zero-divisors and nilpotents
  6. $k$-potents, units, and small elements
  7. Irreducible elements

Key Result

Proposition 1.4

Let $B$ be a content $S$-semialgebra. Then, the content function $c$ from $B$ into $\mathop{\mathrm{Id}}\nolimits(S)$ (see Definition 30 in Nasehpour2016) is an amount function.

Theorems & Definitions (99)

  • Definition 1.1: Amount functions
  • Remark 1.2
  • Proposition 1.4
  • proof
  • Definition 1.5
  • Theorem 1.6
  • proof
  • Remark 1.8
  • Definition 1.9
  • Proposition 1.10
  • ...and 89 more