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arXiv · 1711.05163

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

Yefim Katsov·Tran Giang Nam·Jens Zumbrägel
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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.

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Yefim Katsov, Tran Giang Nam, Jens Zumbrägel. 2017-11-14. On congruence-semisimple semirings and the $K_0$-group characterization of ultramatricial algebras over semifields. https://doi.org/10.1016/j.jalgebra.2018.04.024

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