Maximal orders and reflexive modules

Abstract:

If R is a maximal two-sided order in a semisimple ring and ${M_R}$ is a finite dimensional torsionless faithful R-module, we show that $m = {\text {End}_R}\;{M^\ast }$ is a maximal order. As a consequence, we obtain the equivalence of the following when ${M_R}$ is a generator: 1. M is R-reflexive. 2. $k = {\text {End}}\;{M_R}$ is a maximal order. 3. $k = {\text {End}_R}\;{M^\ast }$ where ${M^\ast } = {\hom _R}(M,R)$. When R is a prime maximal right order, we show that the endomorphism ring of any finite dimensional, reflexive module is a maximal order. We then show by example that R being a maximal order is not a property preserved by k. However, we show that $k = {\text {End}}\;{M_R}$ is a maximal order whenever ${M_R}$ is a maximal uniform right ideal of R, thereby sharpening Faith’s representation theorem for maximal two-sided orders. In the final section, we show by example that even if $R = {\text {End}_k}V$ is a simple pli (pri)-domain, k can have any prescribed right global dimension $\geqslant 1$, can be right but not left Noetherian or neither right nor left Noetherian.