AI 中文总结
研究幂等半环整性的不同定义及关系，通过证明加法幂等半环上的凯莱 - 哈密顿定理作为工具，计算坐标半环及其子半环的整闭包，为热带簇正规化等研究提供途径。
AI 中文摘要
在交换环理论中，整性有多种等价定义。在幂等半环中这些概念有所不同。本文给出了半环整性的不同定义并探究它们之间的关系。作为工具，证明了加法幂等半环上的凯莱 - 哈密顿定理。在示例中，计算了坐标半环在其全分式半环中的整闭包以及坐标半环子半环的整闭包，为定义和理解热带簇的正规化以及热带方式计算簇的正规化提供了途径。
英文摘要
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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