The aim of this paper is to give a new characterization of cyclic modules over local rings, namely, we show that if $N$ is a nonzero finitely generated module over a local (Noetherian) ring $(R,𝔪)$ such that for some integer $n\ge 1$, the submodule ${𝔪}^{n}N$ of $N$ is cyclic, then every submodule of $N$ is cyclic and the ring $R/{\mathrm{\text{Ann}}}_R(N)$ is a $\mathrm{\text{D}}VR$, whenever $\mathrm{\text{depth}}N>0$. As a consequence, it follows that $R/{\mathrm{\text{Ann}}}_R(N/{\mathrm{\Gamma }}_{𝔪}(N))$ is a $\mathrm{\text{D}}VR$ and every submodule of $N/{\mathrm{\Gamma }}_{𝔪}(N)$ is cyclic.

中文翻译：

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局部环上循环模的新表征

本文的目的是给出局部环上循环模块的新特征，即，我们证明如果$ñ$是局部（Noetherian）环上的非零有限生成模块$(R,𝔪)$这样对于某个整数$n\ge 1$, 子模块${𝔪}^nñ$的$ñ$是循环的，那么每个子模$ñ$是循环的并且环$R/{\mathrm{\text{安}}}_R(ñ)$是一个$\mathrm{\text{D}}五R$, 每当$\mathrm{\text{深度}}ñ>0$. 其结果是，$R/{\mathrm{\text{安}}}_R(ñ/{\mathrm{\Gamma }}_{𝔪}(ñ))$是一个$\mathrm{\text{D}}五R$和每个子模块$ñ/{\mathrm{\Gamma }}_{𝔪}(ñ)$是循环的。