Decomposition theories for abelian categories

Both the classical approach to decomposition theories and Fisher's technique of constructing decomposition theories from radical functions are extended to and exploited in the context of abelian categories. These two different approaches to decomposition theories for abelian categories intertwine in one theorem from which flows necessary and sufficient conditions for the existence of the tertiary, primary, and Bourbaki's $\mathcal{P}$-primary decomposition theories.