  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.
