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June 5th, 2011, 04:30 AM   #1
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Finitely generated injective extension

We know that every module has an injective extension; even we have a minimal injective extension known as "injective hall" or injective "envelope".
But this injective extension, even the minimal one, might not be finitely generated. For example consider the Noetherian Z-module Z; one can show that Q as Z-module is its injective hall, which is NOT finitely generated. It also follows that the Z-module Z has no finitely generated injective extension.

So my question is: For which kind of modules, under which conditions, we have a finitely generated injective extension for a module?
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