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.
