The Exact Structure Dervied From A Cotortion Pair

Let(A,?)be an exact category,and let ?(?)A be a subcategory.Let ?? ={0?A?B?C?0(*)exact | for any X ? ?,Hom(X,-)preserves the exactness of(*)}.In Chapter 2,we show that ?? is an exact substructure of(A,?).Let ??-proj={X | X is a ??-projective object},we prove that Add(?)(?)??-proj.In Chapter 3,let(F,C)be a cotorsion pair.We prove that ?F-proj=F;Moreover,if(F,C)is complete,then ?F has enough projective objects.Let ?F={?|? is a ?F-phantom morphism},and let ?F={?|? is a ?F-cophantom morphism}.We show that both ?F and ?F are ideals in A.If A has enough injective objects,and([F,C is a complete cotorsion pair,we prove that ?F:is a special preenveloping ideal.