Ore-Rees rings which are maximal orders

Abstract

Let $R$ be a Noetherian prime ring with an automorphism $\sigma$ and a left $\sigma$-derivation $\delta$, and let $X$ be an invertible ideal of $R$ with $\sigma(X) = X$. We define an Ore-Rees ring $S = R[Xt; \sigma, \delta]$ which is a subring of an Ore extension $R[t; \sigma, \delta]$, where $t$ is an indeterminate. It is shown that if $R$ is a maximal order, then so is $S$. In case $\sigma = 1$, we define the concepts of $(\delta; X)$-stable ideals of $R$ and of $(\delta; X)$-maximal orders and prove that $S$ is a maximal order if and only if $R$ is a $(\delta; X)$-maximal order. Furthermore we give a complete description of v-$S$-ideals, which is used to characterize $S$ to be a generalized Asano ring. In case $\delta = 0$, we define the concepts of $(\sigma; X)$-invariant ideals of $R$ and of $(\sigma; X)$-maximal orders in order to show that $S$ is a maximal order if and only if $R$ is a $(\sigma; X)$-maximal order. We also give examples $R$ such that either $R$ is a $(\delta; X)$-maximal order or is a $(\sigma; X)$-maximal order but they are not maximal orders.