Abstract Let X be a Tychonoff space. Associated with every subset A ⊆ β X is the ideal O A of the ring C ( X ) consisting of all functions that vanish in a neighborhood of X ∩ A . Now, viewing Tychonoff spaces as objects in the category CRLoc of completely regular locales, we have ideals of the form O A , where A is a sublocale of βX. In this paper we study properties of such ideals not only for Tychonoff spaces, but for any object in CRLoc. Carrying out the discussion in this category, we have more function rings (they are denoted R L , for any L ∈ CRLoc ) than the class of the rings C ( X ) . Pure ideals of C ( X ) are known to be exactly the ideals O A , for A a closed subset of βX. We characterize the spaces for which the ideals O A , for A a closed subset of X (note that it need not be closed in βX) are pure ideals. They properly contain the normal ones. We describe the socle of any ring R L as the ideal O A , with A equal to the join of all nowhere dense sublocales of βL. We show that the socle is zero precisely when βL is dense in itself, and essential if and only if the smallest dense sublocale of βL has a complement in the lattice of sublocales of βL. If βL is scattered, then this is so if and only if L has a smallest nowhere dense sublocale.

中文翻译：

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关于与子区域相关的连续函数环的理想

摘要 令 X 为 Tychonoff 空间。与每个子集 A ⊆ β X 相关联的是环 C ( X ) 的理想 OA ，该环由在 X ∩ A 的邻域内消失的所有函数组成。现在，将 Tychonoff 空间视为完全规则区域的类别 CRLoc 中的对象，我们有 OA 形式的理想，其中 A 是 βX 的子区域。在本文中，我们不仅针对 Tychonoff 空间，而且针对 CRLoc 中的任何对象研究此类理想的性质。在这个类别中进行讨论，我们有比环 C ( X ) 的类更多的函数环（它们被表示为 RL ，对于任何 L ∈ CRLoc ）。已知 C ( X ) 的纯理想恰好是理想 OA ，因为 A 是 βX 的封闭子集。我们刻画了理想 OA 的空间，对于 A，X 的封闭子集（注意它不需要在 βX 中封闭）是纯理想。它们正确地包含了正常的。我们将任何环 RL 的 socle 描述为理想的 OA ，其中 A 等于 βL 的所有无处密集子区域的连接。我们表明，当 βL 本身是稠密的时，socle 恰好为零，并且当且仅当 βL 的最小稠密子区域在 βL 的子区域的格子中具有补集时才是必要的。如果 βL 是分散的，那么当且仅当 L 具有最小的无处稠密子区域。