Ore-Rees rings which are maximal orders

Abstract

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