Nearring 𝑝-Indecomposables and Φ-Irreducibility

This paper is mainly about direct summand right ideals of nearrings with 𝐷𝐶𝐶𝑁 which cannot be expressed as a non-trivial direct sum. A fairly natural condition (Φ-irreducibility) makes it possible to study these right ideals in reasonable depth. It turns out they are either very ring like or right ideals (called shares) controlling considerable nearring structure. The two cases are studied in some detail. A surprising feature of the last section is that, with weak hypercentrality present, the nearring is a unique finite direct sum of these right ideals if, and only if, all such right ideals are ideals.