Every simple compact semiring is finite

TL;DR

This paper proves that all simple compact semirings are finite, extending classical results from rings to semirings, and showing that infinite compact semirings have non-trivial quotients.

Contribution

It generalizes the classical finiteness result from simple compact rings to simple compact semirings, broadening the scope of algebraic structure classification.

Findings

Every simple compact semiring is finite.

Infinite compact semirings have proper non-trivial quotients.

Classical results for rings are extended to semirings.

Abstract

A Hausdorff topological semiring is called simple if every non-zero continuous homomorphism into another Hausdorff topological semiring is injective. Classical work by Anzai and Kaplansky implies that any simple compact ring is finite. We generalize this result by proving that every simple compact semiring is finite, i.e., every infinite compact semiring admits a proper non-trivial quotient.