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Endomorphism rings of torsionless modules. (English) Zbl 0224.16005
Let $$A$$ be a ring, $$M_A$$ be a right $$A$$-module and $$B=\text{End}\,M_A$$. The usual Morita theorems establish a close connection between $$A$$-submodules of $$M_A$$ and right ideals of $$B$$ when $$M_A$$ is a progenerator in mod-$$A$$. Recently, C. Faith has shown [Bull. Am. Math. Soc. 77, 338–342 (1971; Zbl 0217.06002)] that this connection can be retained in a suitably modified form when $$M_A$$ is a generator in mod-$$A$$. The present paper uses these ideas to prove the following theorem of J. M. Zelmanowitz [J. Algebra 5, 325–341 (1967; Zbl 0155.36102)]:
“Let $$A$$ be a right order in a semisimple ring $$\Sigma$$, $$M_A$$ be a finite dimensional torsion-less right $$A$$-module, $$E(M_A)$$ be the injective hull of $$M_$$A, $$B=\text{End}\,M_A$$ and $$Q=\text{End}\,E(M_A)$$. Then $$B$$ is a right order in the semisimple ring $$Q$$. If $$A$$ is a two-sided order in $$E$$ then $$B$$ is a two-sided order in $$Q$$”.
We also obtain a new related result viz., “If $$A$$ is bounded then so is $$B$$”. The paper ends with an example which may be of some interest. We construct a two-sided noetherian prime ring $$A$$ and a right ideal $$M_A$$ of $$A$$ such that $$\text{End}\,M_A$$ is not noetherian from either side. This example can be modified to obtain a related example for artinian rings.
Reviewer: A. V. Jategaonkar

##### MSC:
 16D10 General module theory in associative algebras 16D25 Ideals in associative algebras 16P20 Artinian rings and modules (associative rings and algebras)
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##### References:
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