Congruence-simple multiplicatively idempotent semirings

Tomáš Kepka; Miroslav Korbelář; Günter Landsmann

Congruence-simple multiplicatively idempotent semirings

Rings and Algebras 2022-07-19 v1

Authors: Tomáš Kepka , Miroslav Korbelář , Günter Landsmann

Abstract

Let $S$ be a multiplicatively idempotent congruence-simple semiring. We show that $|S|=2$ if $S$ has a multiplicatively absorbing element. We also prove that if $S$ is finite then either $|S|=2$ or $S\cong End(L)$ or $S^{op}\cong End(L)$ where $L$ is a 2-element semilattice. It seems to be an open question, whether $S$ can be infinite at all.

Keywords

commutative ring theory lattice theory finite group theory

Cite

@article{arxiv.2207.08160,
  title  = {Congruence-simple multiplicatively idempotent semirings},
  author = {Tomáš Kepka and Miroslav Korbelář and Günter Landsmann},
  journal= {arXiv preprint arXiv:2207.08160},
  year   = {2022}
}

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