Contracted Ideals of $p$-adic Integral Group Rings

Abstract

Let $G$ be a cyclic group of order $p^k$ for a prime $p$. There is a natural embedding of $\mathbb{Z}_p[G]$ into a product of rings of integers of cyclotomic fields. We consider the ideals $I$ of $\mathbb{Z}_p[G]$ for which the following statement holds: if $\alpha\in\mathbb{Z}_p[G]$ and $\chi(\alpha)\in\chi(I)$ for all $p$-adic characters $\chi$ of $G$, then $\alpha\in I$. We show that they are precisely the contracted ideals of the embedding. Powers of the augmentation ideal of $\mathbb{Z}_p[G]$ are not contracted in general, and we calculate the obstruction of contraction in this case.