Let A be a right order in a semisimple ring
Σ
,
M
A
\Sigma ,{M_A}
be a finite-dimensional torsionless right A-module and
M
^
A
{\hat M_A}
be the injective hull of M. J. M. Zelmanowitz has shown that
Q
=
E
n
d
M
^
A
Q = {\rm {End}}\;{\hat M_A}
is a semisimple ring and
S
=
E
n
d
M
A
S = {\rm {End}}\;{M_A}
is a right order in Q. Further, if A is a two-sided order in
Σ
\Sigma
then S is a two-sided order in Q. We give a conceptual proof of this result. Moreover, we show that if A is a bounded order then so is S. The underlying idea of our proofs is very simple. Rather than attacking
S
=
E
n
d
M
A
S = {\rm {End}}\;{M_A}
directly, we prove the results for
B
=
E
n
d
(
M
A
⊕
A
A
)
B = {\rm {End}}\;({M_A} \oplus {A_A})
. If
e
:
M
A
⊕
A
A
→
M
A
⊕
A
A
e:{M_A} \oplus {A_A} \to {M_A} \oplus {A_A}
is the projection on M along
A
A
{A_A}
then, of course,
S
≅
e
B
e
S \cong eBe
and it is easy to transfer the required information from B to S. The reason why it is any easier to look at B rather than S is that
M
A
⊕
A
A
{M_A} \oplus {A_A}
is a generator in
mod
-
A
\bmod \text {-}A
and a Morita type transfer of properties from A to B is available. We construct an Artinian ring resp. Noetherian prime ring containing a right ideal whose endomorphism ring fails to be Artinian resp. Noetherian from either side.