We show that ideal submodules and closed ternary ideals in Hilbert modules are the same. We use this insight as a little peg on which to hang a little note about interrelations with other notions regarding Hilbert modules. In Section 3, we show that the ternary ideals (and equivalent notions) merit fully, in terms of homomorphisms and quotients, to be called ideals of (not necessarily full) Hilbert modules. The properties to be checked are intrinsically formulated for the modules (without any reference to the algebra over which they are modules) in terms of their ternary structure. The proofs, instead, are motivated from a third equivalent notion, linking ideals (Section 2), and a Theorem (Section 3) that all extends nicely to (reduced) linking algebras. As an application, in Section 4, we introduce ternary extensions of Hilbert modules and prove most of the basic properties (some new even for the known notion of extensions of Hilbert modules), by reducing their proof to the well-known analogue theorems about extensions of C^∗–algebras. Finally, in Section 5, we propose several open problems that our method naturally suggests.

我们证明了 Hilbert 模中的理想子模和闭三元理想是相同的。我们使用这种见解作为一个小钉子，在上面挂一点关于与其他关于 Hilbert 模块的概念的相互关系的注释。在第 3 节中，我们表明三元理想（和等价概念）在同态和商方面完全值得称为（不一定是完整的）希尔伯特模的理想。要检查的属性是根据模块的三元结构本质上为模块制定的（没有任何参考它们是模块的代数）。相反，证明来自第三个等效概念，即连接理想（第 2 节）和定理（第 3 节），它们都很好地扩展到（简化的）连接代数。作为应用程序，在第 4 节中，C ^{* -}代数。最后，在第 5 节中，我们提出了我们的方法自然建议的几个开放问题。