Endomorphism rings of torsionless modules

Abstract:

Let A be a right order in a semisimple ring $\Sigma ,{M_A}$ be a finite-dimensional torsionless right A-module and ${\hat M_A}$ be the injective hull of M. J. M. Zelmanowitz has shown that $Q = {\rm {End}}\;{\hat M_A}$ is a semisimple ring and $S = {\rm {End}}\;{M_A}$ is a right order in Q. Further, if A is a two-sided order in $\Sigma$ 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 $S = {\rm {End}}\;{M_A}$ directly, we prove the results for $B = {\rm {End}}\;({M_A} \oplus {A_A})$. If $e:{M_A} \oplus {A_A} \to {M_A} \oplus {A_A}$ is the projection on M along ${A_A}$ then, of course, $S \cong eBe$ 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 ${M_A} \oplus {A_A}$ is a generator in $\bmod \text {-}A$ 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.