Endomorphism rings of torsionless modules

Author:
Arun Vinayak Jategaonkar

Journal:
Trans. Amer. Math. Soc. **161** (1971), 457-466

MSC:
Primary 16.40

DOI:
https://doi.org/10.1090/S0002-9947-1971-0284464-9

MathSciNet review:
0284464

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Abstract: Let *A* be a right order in a semisimple ring be a finite-dimensional torsionless right *A*-module and be the injective hull of *M*. J. M. Zelmanowitz has shown that is a semisimple ring and is a right order in *Q*. Further, if *A* is a two-sided order in 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 directly, we prove the results for . If is the projection on *M* along then, of course, 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 is a generator in - 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0284464-9

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