Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16.40

Retrieve articles in all journals with MSC: 16.40


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0284464-9
PII: S 0002-9947(1971)0284464-9
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