On congruence-semisimple semirings and the $K_0$-group characterization of ultramatricial algebras over semifields

TL;DR

This paper characterizes congruence-semisimple semirings and demonstrates that $K_0$-groups serve as complete invariants for ultramatricial algebras over semifields, extending classification results in semiring theory.

Contribution

It provides a full description of congruence-semisimple semirings and establishes $K_0$-groups as complete invariants for ultramatricial algebras over semifields.

Findings

$K_0$-groups classify ultramatricial algebras over semifields.

$SK_0$-groups characterize zerosumfree congruence-semisimple semirings.

Complete invariants for certain semirings are established.

Abstract

In this paper, we provide a complete description of congruence-semisimple semirings and introduce the pre-ordered abelian Grothendieck groups $K_0(S)$ and $SK_0(S)$ of the isomorphism classes of the finitely generated projective and strongly projective S-semimodules, respectively, over an arbitrary semiring S. We prove that the $SK_0$-groups and $K_0$-groups are complete invariants of, i.e., completely classify, ultramatricial algebras over a semifield F. Consequently, we show that the $SK_0$-groups completely characterize zerosumfree congruence-semisimple semirings.