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June 5th, 2011, 03:30 AM  #1 
Member Joined: Aug 2010 Posts: 32 Thanks: 0  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 Zmodule Z; one can show that Q as Zmodule is its injective hall, which is NOT finitely generated. It also follows that the Zmodule 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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extension, finitely, generated, injective 
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