In this paper, we describe how intersections with a totality of some ideals affect the essentiality of an ideal. We mainly study intersections with every (a) annihilator ideal, (b) prime ideal (c) strongly irreducible ideal (d) irreducible ideal and every pure ideal. After some general results, the paper focuses on C(X) to characterize spaces X when every irreducible ideal of C(X) is pseudoprime. We also characterize the rings of continuous functions C(X) in which every pseudoprime ideal is strongly irreducible. We give a negative answer to a question raised by Gilmer and McAdam.

中文翻译：

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交换环中理想的本质性变体

在本文中，我们描述了与一些理想的整体的交集如何影响理想的本质。我们主要研究与每个 (a) 湮没理想、(b) 素理想 (c) 强不可约理想 (d) 不可约理想和每个纯理想的交集。在得到一些一般性的结果之后，本文着重于C(X)表征空间X当每一个不可约的理想C(X)是伪素数。我们还刻画了连续函数的环C(X)其中每个伪素理想都是强不可约的。我们对 Gilmer 和 McAdam 提出的问题给出否定的回答。