A Course in Homological Algebra by P. J. Hilton, U. Stammbach (auth.)

By P. J. Hilton, U. Stammbach (auth.)

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By Zorn's Lemma there exists a maximal essential extension E of A which is contained in M. 2. Let A be a submodule of the injective module I. Let E be a maximal essential extension of A contained in I. Then E is injective. Proof. First we show that E does not admit any non-trivial essential monomorphism. Let 11: E-+X be an essential monomorphism. / E~X ai········ I We show that ~ is monomorphic. Let H be the kernel of ~. We then have H <;; X and H n IlE = O. Hence ker ~ = H = 0, for 11 is essential.

Using the right A-module structure of A we define in Homz(A, G) a left A-module structure as follows: (Acp)(a)=cp(a},,), aEA, AEA, cpEHomz(A,G). We leave it to the reader to verify the axioms. Similarly if A is a left A-module, Homz(A, G) acquires the structure of a right A-module. 1. Let A be a left A-module and let G be an abelian group. Regard Homz(A, G) as a left A-module via the right A-module structure of A. Then there is an isomorphism of abelian groups '1 ='1A: HomA(A, Homz(A, G))-"4Homz(A, G).

1. e. 2) IX I: =/3I:=1X =/3. 1. 2). It is then plain that, if cp is a morphism in (t, then cp is a monomorphism in (t if and only if it is an epimorphism as a morphism of (t0pp. 3) that a statement about epimorphisms and monomorphisms which is true in any category must remain true if the prefixes "epi-" and "mono-" are interchanged and "arrows are reversed". Let us take a trivial example. An easy argument establishes the fact that if cptp is monomorphic then tp is monomorphic. We may thus apply the "duality principle" to infer immediately that if tpcp is epimorphic then tp is epimorphic.

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