发表机构
University of Engineering and Technology, Vietnam National University; Department of Mathematics, School of Science and Technology, Meiji University（越南国立大学工程技术大学; 明治大学理工学部数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究一维Cohen-Macaulay局部环的典范理想幂的迹，证明几乎Gorenstein性质等价于σ(R)≥v，并刻画σ(R)达到最大有限值v-1的环结构，应用于数值半群环。
AI 中文摘要
设 $(R,\m)$ 是一维 Cohen--Macaulay 局部环，具有典范分式理想 $R\subseteq K\subseteq \overline R$，并设 $v$ 表示其嵌入维数。令 $\sigma(R)=\sup\{j\ge0\mid\tr(K^j)\supseteq\m\}$。根据定义，$R$ 是近 Gorenstein 环当且仅当 $\sigma(R)>0$。我们证明 $R$ 是几乎 Gorenstein 环当且仅当 $\sigma(R)\ge v$，等价地，$\sigma(R)=\infty$。因此 $\sigma(R)$ 的每个有限值至多为 $v-1$。然后我们确定 $\sigma(R)$ 达到其最大有限值 $v-1$（刚好低于几乎 Gorenstein 性质的阈值）的环的结构。这当且仅当 $K^{v-1}\cong\m$ 且 $K^v\not\cong\m$ 时发生。当这些条件成立时，$R$ 具有类型二，重数 $e$ 大于 $v$ 且 $e \equiv 3 \pmod{v-1}$。作为应用，我们将满足 $\sigma(R)=v-1$ 的数值半群环 $R$ 刻画为那些由算术级数生成、类型二且重数大于 $v$ 的环。
英文摘要
Let $(R,\m)$ be a one-dimensional Cohen--Macaulay local ring with a canonical fractional ideal $R\subseteq K\subseteq\overline R$, and let $v$ denote its embedding dimension. Put $σ(R)=\sup\{j\ge0\mid\tr(K^j)\supseteq\m\}$. By definition, $R$ is nearly Gorenstein if and only if $σ(R)>0$. We prove that $R$ is almost Gorenstein if and only if $σ(R)\ge v$, equivalently, $σ(R)=\infty$. Thus every finite value of $σ(R)$ is at most $v-1$. We then determine the structure of rings for which $σ(R)$ attains its largest finite value $v-1$, just below the threshold for the almost Gorenstein property. This occurs if and only if $K^{v-1}\cong\m$ and $K^v\not\cong\m$. When these conditions hold, $R$ has type two, with multiplicity $e$ greater than $v$ and $e \equiv 3 \pmod{v-1}$. As an application, we characterize the numerical semigroup rings $R$ with $σ(R)=v-1$ as those generated by an arithmetic progression, of type two and multiplicity greater than $v$.
