One may often decompose the domain of a topologically transitivemap into finitely many regular closed pieces with nowhere denseoverlap in such a way that these pieces map into one another in aperiodic fashion. We call decompositions of this kind regularperiodic decompositions and refer to the number of pieces asthe length of the decomposition. If $f$ is topologically transitivebut $f^{n}$ is not, then $f$ has a regular periodic decomposition ofsome length dividing $n$. Although a decomposition of a givenlength is unique, a map may have many decompositions ofdifferent lengths. The set of lengths of decompositions of a givenmap is an ideal in the lattice of natural numbers ordered bydivisibility, which we call the decomposition ideal of $f$. Everyideal in this lattice arises as a decomposition ideal of some map.Decomposition ideals of Cartesian products of transitive mapsare discussed and used to develop various examples. Results areobtained concerning the implications of local connectedness fordecompositions. We conclude with a comprehensive analysis ofthe possible decomposition ideals for maps on 1-manifolds.