By Leeb B.

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Let E be an inJective mcdule. Consider all families (Ei) of indecomposable injective submodules such that the su~ ~E i in E is direct. By Zorn,s lemma there exists a maximal such family (Ei) I . 2 we know that the sum injective module, (D E i is an I E = (v~Ei) (D E' . We want to sc we can write ~M show that E' = 0 , and fcr this it suffices tc show that every injeotive module Let x E' \$ 0 be any element modules E" of E contains an indecomposable direct summand. • 0 in such that E' . Consider all injective subx ~ E" .

The exact sequence Hom(F,E)---, ~om(K,E) ;ExtZ~M,E) together with (c), shows that (b) =~(a) An ExtI(M,E) = 0 . 3. The su~od~e is an , is trivial. Definition. M C, E , 0 ~-injective envelope of M is a monomorphism M ¢ ~(E) . {x ~ E(M) I (,,x) c ~ } of E(M) M . Proof. Clearly the subset in question is a suhmodule of It suffices to show that Suppose we are given E' = { x C E(M)~ (M:x) C ~ } is f:l-~E' extended %o a homomorphism with g:A-*E(M) I ¢ F . f E(M) . [-injecti,. , If A E'~E(M) L E,/~ ~ h % 0 , then t E(~)/M ~ E(M)/E' : A/I ~ 0 ;E(~)/E, , 0 , 0 U E(,)/E, contains a non-zero torsion submodule, and from the lowest row we see that this contradicts the fact that 3!

MA is F-olosed if the oanonioal maps M ~ Ho~A(A,M ) ~ are isomorphisms for all Thus and M M is HomA(I,M ) I c F . F-closed if and only if M is both torsion-free ~-injective (as defined in § 6). 8. MF that if M x e M(F_) T M Z M ~---~MF . Conversely we have: is Proof. To show that F-closed for every module MF is torsion-free, is torsion-free, and xJ = 0 F__-elosed module for then some M(_F) MA . it suffices to show is torsion-free. J ¢ F_ . L e t x Suppose be represented :I - ~ M . 4 we have a commutative diagram I ( r A = xa M so ~M~ so Next we show that with H/t(xl J ~|INJ MF = 0 is and If~J s F .