On congruence-semisimple semirings and the K0-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 K0(S) and SK0(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 SK0-groups and K0-groups are complete invariants of, i.e., completely classify, ultramatricial algebras over a semifield F. Consequently, we show that the SK0-groups completely characterize zerosumfree congruence-semisimple semirings.