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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

16D10 General module theory in associative algebras
16D25 Ideals in associative algebras
16P20 Artinian rings and modules (associative rings and algebras)
Full Text: DOI
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