The purpose of this paper is, as part of the stratification of Cohen-Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the fiber product $R \times_T S$ of Cohen-Macaulay local rings $R$, $S$ of the same dimension $d>0$ over a regular local ring $T$ with $\dim T=d-1$ is an almost Gorenstein ring if and only if so are $R$ and $S$. Besides, the other generalizations of Gorenstein properties are also explored.

中文翻译：

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几乎来自纤维产品的 GORENSTEIN 戒指

本文的目的是，作为 Cohen-Macaulay 环分层的一部分，研究何时纤维产品几乎是 Gorenstein 环的问题。我们证明了 Cohen-Macaulay 局部环 $R$ 的纤维积 $R\times_T S$，相同维度 $d>0$ 的 $S$ 超过了常规局部环 $T$，$\dim T=d- 1$ 几乎是 Gorenstein 环当且仅当 $R$ 和 $S$ 是。此外，还探讨了 Gorenstein 性质的其他概括。