On p-modular system (K, O, k) - Joshua (2019)
This objective of this report is to summarise several concepts about the relation between idempotents in p-modular system (K, O, k) and how it can help reader in understanding the relation between structure of ring and its modulo and also provide reader with some application related to this concepts. The important theorem in this report would be Theorem 4.3.2 that tells us about the dependency of existence among idempotent in a ring and its modulo, Lemma 4.2.4 that will help us to decompose a ring into direct sum of ideals using a complete set of orthogonal idempotents of a ring, and lastly Theorem 4.3.6 that describe the existence of isomorphic map from any kG-module to a module of the form U for some A-free AG-module U. Utilising these theorems, we are able to apply it to find a complete set of orthogonal idempotent in group algebra (Z/pZ)Cn and (Zp/pZp)Cn and use the result to deduce the relation between idempotent and structure in both group algebra.