Abstract

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