Hemiring-valued pseudonormed rings

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.