Decompositions of injective modules

Abstract

The main results of this paper concern decompositions of an injective module, either as a direct sum of submodules or as the injective envelope of a direct sum of injective submodules. This second kind of decomposition can be regarded as an ordinary direct sum (coproduct) in a suitable Abelian category-the spectral category of the ring. The results are therefore put in the context of Abelian categories, and the main result is that in an Abelian category satisfying axiom Ab-5 and with infinite direct sums, any two direct sum decompositions of an injective object have isomorphic refinements. This is particularly strong if decompositions into indecomposable injectives exist, and it enables one to classify the injective modules over a valuation ring. Such strong results as this are not available for more general classes of modules, but in 3 the methods of Crawley and Jnsson are exploited to obtain results in certain cases; for example, for modules which are direct sums of countably generated modules. The Crawley-Jnsson results are put into the context of category theory and an example is given (involving relatively injective modules) to show how the hypotheses can be weakened by working in a subcategory of the category of j-modules.