Abstract Let R be commutative ring with . A proper ideal I of R is called a 1-absorbing primary ideal of R if whenever nonunit elements and , then or . It is proved that every primary ideal of R is 1-absorbing primary and every 1-absorbing primary ideal of R is semi-primary (that is ideals with prime radical). However, these three concepts are different. In this paper, we characterize rings R over which every semi-primary ideal is 1-absorbing primary and (resp. Noetherian) rings R over which every 1-absorbing primary ideal is prime (resp. primary). Many examples are given to illustrate the obtained results.

中文翻译：

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每个半初级理想都是 1 吸收初级的环

摘要 令 R 是与 的交换环。R 的适当理想 I 称为 R 的 1 吸收初级理想，如果每当非单位元素和 ，则 或 。证明了R的每一个初理想都是吸1的初理想，R的每一个吸1的初理想都是半初理想（即有素根的理想）。但是，这三个概念是不同的。在本文中，我们描述了环 R ，其中每个半主理想都是 1 吸收主理想，以及（相应的诺特）环 R，其中每个 1 吸收主理想都是素数（相应主理想）。给出了许多例子来说明得到的结果。