Hemiring-valued pseudonormed rings

TL;DR

This paper introduces hemiring-valued pseudonormed rings, generalizes norming finite-dimensional algebras, and explores properties of shrinkable hemirings, dense division semirings, and normed groups with new convergence tests.

Contribution

It extends the concept of normed rings to hemiring-valued structures and proves new properties and tests related to these generalized algebraic systems.

Findings

Generalization of Albert's result to hemiring-valued pseudonormed rings

Introduction of shrinkable hemirings and their properties

Establishment of convergence tests for ring-valued normed groups

Abstract

In the second section, we introduce hemiring-valued pseudonormed rings and generalize Albert's result which states that every finite-dimensional algebra can be normed. Next, we introduce shrinkable hemirings and prove that dense division semirings are shrinkable. In the third section, we show the Cauchy Condensation Test holds for Cauchy complete fields. In the fourth section, we use Bernoulli's inequality to prove a version of ratio test for ring-valued normed groups.