Semirings generated by idempotents
Rings and Algebras
2024-04-12 v1
Abstract
We prove that a semiring multiplicatively generated by its idempotents is commutative and Boolean, if every idempotent in the semiring has an orthogonal complement. We prove that a semiring additively generated by its idempotents is commutative, if every idempotent in the semiring has an orthogonal complement and all the nilpotents in the semirings are central. We also provide examples that the assumptions on the existence of orthogonal complements of idempotents and the centrality of nilpotents cannot be omitted.
Keywords
Cite
@article{arxiv.2404.07623,
title = {Semirings generated by idempotents},
author = {David Dolžan},
journal= {arXiv preprint arXiv:2404.07623},
year = {2024}
}