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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Endomorphism rings of torsionless modules

Author: Arun Vinayak Jategaonkar
Journal: Trans. Amer. Math. Soc. 161 (1971), 457-466
MSC: Primary 16.40
MathSciNet review: 0284464
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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$-$ 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.

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Keywords: Finite-dimensional modules, torsionless modules, nonsingular modules, generators, endomorphism rings, orders in semisimple rings, bounded orders, Morita theorems, prime Noetherian rings, Artinian ring
Article copyright: © Copyright 1971 American Mathematical Society