We show that Ringrose's diagonal ideals are primitive ideals in a nest algebra (subject to the continuum hypothesis). This answers an old question of Lance and provides for the first time concrete descriptions of enough primitive ideals to obtain the Jacobson radical as their intersection. Separately, we provide a standard form for all left ideals of a nest algebra, which leads to insights into the maximal left ideals. In the case of atomic nest algebras, we show how primitive ideals can be categorized by their behaviour on the diagonal and provide concrete examples of all types.

中文翻译：

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论巢代数的原始理想

我们证明 Ringrose 的对角理想是巢代数中的原始理想（服从连续统假设）。这回答了 Lance 的一个老问题，并首次提供了对足够原始理想的具体描述，以获得 Jacobson 激进作为它们的交集。另外，我们为嵌套代数的所有左理想提供了一个标准形式，从而可以深入了解最大左理想。在原子嵌套代数的情况下，我们展示了原始理想如何通过它们在对角线上的行为进行分类，并提供所有类型的具体示例。