AI 中文总结
本文引入有限范畴$\mathcal{C}$的简单伯恩赛德环$B^S(\mathcal{C})$，它由简单$\mathcal{C}$-集构造，推广了有限群的经典伯恩赛德环且秩有限。研究其基础理论与结构性质，包括确定到$\mathbb{Z}$的同态、描述素谱及得到分解定理。
AI 中文摘要
由韦伯引入的有限范畴的伯恩赛德环推广了有限群的经典伯恩赛德环。但与经典情形不同，有限范畴的伯恩赛德环当且仅当该范畴等价于一个广群时才有有限秩。本文引入了与有限范畴$\mathcal{C}$相关的新不变量，即$\mathcal{C}$的简单伯恩赛德环，记为$B^S(\mathcal{C})$。它由简单$\mathcal{C}$-集构造而来，推广了有限群的经典伯恩赛德环，且总是具有有限秩。我们发展了简单$\mathcal{C}$-集的基础理论并研究了$B^S(\mathcal{C})$环的若干结构性质。特别地，确定了从$B^S(\mathcal{C})$到$\mathbb{Z}$的所有环同态，描述了其素谱，并得到了将$B^S(\mathcal{C})$表示为强连通子范畴的简单伯恩赛德环之积的分解定理。
英文摘要
The Burnside ring of a finite category, introduced by Webb, generalizes the classical Burnside ring of a finite group. However, unlike the classical case, the Burnside ring of a finite category has finite rank if and only if the category is equivalent to a groupoid. In this article, we introduce a new invariant associated with a finite category $\mathcal{C}$, called the \emph{simple Burnside ring} of $\mathcal{C}$ and denoted by $B^S(\mathcal{C})$. This construction is obtained from simple $\mathcal{C}$-sets and generalizes the classical Burnside ring of a finite group. Moreover, the ring $B^S(\mathcal{C})$ always has finite rank. We develop the basic theory of simple $\mathcal{C}$-sets and study several structural properties of the ring $B^S(\mathcal{C})$. In particular, we determine all ring homomorphisms from $B^S(\mathcal{C})$ to $\mathbb{Z}$, describe its prime spectrum, and obtain a decomposition theorem expressing $B^S(\mathcal{C})$ as a product of simple Burnside rings of strongly connected subcategories.
