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

In this paper, we provide a complete description of congruence-semisimple semirings and introduce the pre-ordered abelian Grothendieck groups $K_0(S)$ and $S{K}_0(S)$ of the isomorphism classes of the finitely generated projective and strongly projective S-semimodules, respectively, over an arbitrary semiring S. We prove that the $S{K}_0$-groups and $K_0$-groups are complete invariants of, i.e., completely classify, ultramatricial algebras over a semifield F. Consequently, we show that the $S{K}_0$-groups completely characterize zerosumfree congruence-semisimple semirings.