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We show that ideal submodules and closed ternary ideals in Hilbert modulesare the same. We use this insight as a little peg on which to hang a littlenote about interrelations with other notions regarding Hilbert modules. InSection 3, we show that the ternary ideals (and equivalent notions) meritfully, in terms of homomorphisms and quotients, to be called ideals of (notnecessarily full) Hilbert modules. The properties to be checked areintrinsically formulated for the modules (without any reference to the algebraover which they are modules) in terms of their ternary structure. The proofs,instead, are motivated from a third equivalent notion, linking ideals (Section2), and a Theorem (Section 3) that all extends nicely to (reduced) linkingalgebras. As an application, in Section 4, we introduce ternary extensions ofHilbert modules and prove most of the basic properties (some new even for theknown notion of extensions of Hilbert modules), by reducing their proof to thewell-known analogue theorems about extensions of $C^*$-algebras. Finally, inSection 5, we propose several open problems that our method naturally suggests.

Ideal Submodules versus Ternary Ideals versus Linking Ideals