Abstract Let R be a commutative ring with identity and S be a multiplicative subset of R. In this paper, we introduce the concept of weakly S-prime ideals which is a generalization of weakly prime ideals. Let P be an ideal of R disjoint with S. We say that P is a weakly S-prime ideal of R if there exists an s ∈ S such that, for all a, b ∈ R, if 0 ≠ ab ∈ P, then sa ∈ P or sb ∈ P. We show that weakly S-prime ideals have many analog properties to these of weakly prime ideals. We also use this new class of ideals to characterize S-Noetherian rings and S-principal ideal rings.

中文翻译：

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关于交换环的弱S-素理想

摘要 设R是一个具有恒等式的交换环，S是R的一个乘法子集。在本文中，我们介绍了弱S-素理想的概念，它是弱素理想的推广。令 P 是 R 与 S 不相交的理想。我们说 P 是 R 的弱 S-素理想如果存在一个 s ∈ S 使得对于所有 a, b ∈ R, 如果 0 ≠ ab ∈ P, 那么sa ∈ P 或 sb ∈ P。我们证明了弱 S-素数理想与弱素数理想有许多类似的性质。我们还使用这一类新的理想来表征 S-Noetherian 环和 S-主要理想环。