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

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

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Let 0-+ R -+ F -+ A -+0 be a short exact sequence of abelian groups, with F free. By embedding F in a direct sum of copies of

Let F, G be two functors from the category (t to the category 1). Then a natural transformation t from F to G is a rule assigning to each object X in (t a morphism tx: FX -GX in 1) such that, for any morphism f: X - Y in (t, the diagram FX~GX Ffl 1Gf FY~GY commutes. If tx is isomorphic for each X then t is called a natural equivalence and we write F~ G. It is plain that then t- 1 : G~F, where t- 1 is given by (t- 1h = (tX)-I. If t: F -G, u: G-H are natural transformations then we may form the composition ut: F - H, given by (uth = (u x ) (tx); and the composition of natural transformations is plainly associative.

1. Let P = E8 A j , where Aj = A, be a free module jeJ and let R be a submodule of P. We shall show that R has a basis. Assume J well-ordered and define for every j E J modules ~j)= ffiAi' i-+P(j)nR-imfj . Clearly imfj is an ideal in A. Since A is a principal ideal domain, this ideal is generated by one element, say Aj .

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