Reflexive Ideals in Iwasawa Algebras

Details Reflexive Ideals in Iwasawa Algebras Reflexive Ideals in Iwasawa Algebras

2007-10-02

arxiv.org/abs/0710.0624v1

Mathematics

Rings and Algebras

Scientific paper

Let $G$ be a torsionfree compact $p$-adic analytic group. We give sufficient conditions on $p$ and $G$ which ensure that the Iwasawa algebra $\Omega_G$ of $G$ has no non-trivial two-sided reflexive ideals. Consequently, these conditions imply that every nonzero normal element in $\Omega_G$ is a unit. We show that these conditions hold in the case when $G$ is an open subgroup of $\SL_2(\Zp)$ and $p$ is arbitrary. Using a previous result of the first author, we show that there are only two prime ideals in $\Omega_G$ when $G$ is a congruence subgroup of $\SL_2(\Zp)$: the zero ideal and the unique maximal ideal. These statements partially answer some questions asked by the first author and Brown.

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