A Course in Homological Algebra by Peter J. Hilton, Urs Stammbach

I ~ E now splits by the embedding of E in I. 3. 3. Let E 1 , E2 be two maximal essential extensions of A contained in injective modules II' 12, Then El ~ E2 and every injective module I containing A also contains a submodule isomorphic to El . Definition. El is called the injective envelope of A. Proof. Consider the diagram r~E' E2 Since E2 is injective there exists ~: El ~ E2 completing the diagram. 2 one shows that ~ is monomorphic. 2, that ~: El ~ E 2ยท The proof of the second part is now trivial.

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