Integral closure for (additively idempotent) semirings
arXiv:2607.11219
The paper studies various notions of integrality for (additively idempotent) semirings, compares them, and proves a Cayley‑Hamilton theorem in this setting, with applications to computing integral closures and normalizations of tropical varieties.
Abstract
In commutative ring theory there are multiple equivalent definitions of integrality. These notions diverge when working with idempotent semirings. In this paper we present these different definitions of integrality for semirings and explore the relations between them. As a tool, we prove a Cayley-Hamilton theorem over additively idempotent semirings, which may be of broader interest. In examples, we compute integral closures of coordinate semirings in their total semiring of fractions and integral closures of sub-semirings of coordinate semirings. Such computation gives avenues to defining and understanding the normalization of tropical varieties as well as computing normalization of varieties tropically.
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