Maximal orders and reflexive modules

Abstract

If R is a maximal two-sided order in a semisimple ring and M R {M_R} is a finite dimensional torsionless faithful R-module, we show that m = End R M ∗ m = {\text {End}_R}\;{M^\ast } is a maximal order. As a consequence, we obtain the equivalence of the following when M R {M_R} is a generator: 1. M is R-reflexive. 2. k = End M R k = {\text {End}}\;{M_R} is a maximal order. 3. k = End R M ∗ k = {\text {End}_R}\;{M^\ast } where M ∗ = hom R ( M , R ) {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 = End M R k = {\text {End}}\;{M_R} is a maximal order whenever M R {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 = End k V R = {\text {End}_k}V is a simple pli (pri)-domain, k can have any prescribed right global dimension ⩾ 1 \geqslant 1 , can be right but not left Noetherian or neither right nor left Noetherian.