Recent progress has been done recently in the study of direct-sum decompositions
in the category of modules. We will try to extend these results to the setting
of arbitrary additive categories, or, better to additive categories in which
idempotents have kernels, equivalently, categories in which idempotents split.
For these categories, called amenable by Freyd, the Krull-Schmidt Theorem
holds for objects with a local endomorphism ring, as was observed by Bass. Our
main ingredients will be valuations of additive categories and categories with
invariant basis number.