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.