# zbMATH — the first resource for mathematics

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
Show Scanned Page ##### MSC:
 16D10 General module theory in associative algebras 16D25 Ideals in associative algebras 16P20 Artinian rings and modules (associative rings and algebras)
Full Text:
##### References:
  S. A. Amitsur, Rings of quotients and Morita contexts, J. Algebra 17 (1971), 273 – 298. · Zbl 0221.16014 · doi:10.1016/0021-8693(71)90034-2 · doi.org  Hyman Bass, Algebraic \?-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. · Zbl 0174.30302  Carl Faith, A correspondence theorem for projective modules and the structure of simple noetherian rings, Bull. Amer. Math. Soc. 77 (1971), 338 – 342. · Zbl 0217.06002  R. Hart, Endomorphisms of modules over semi-prime rings, J. Algebra 4 (1966), 46 – 51. · Zbl 0145.04302 · doi:10.1016/0021-8693(66)90049-4 · doi.org  Nathan Jacobson, The Theory of Rings, American Mathematical Society Mathematical Surveys, vol. II, American Mathematical Society, New York, 1943. · Zbl 0060.07302  Arun Vinayak Jategaonkar, A counter-example in ring theory and homological algebra, J. Algebra 12 (1969), 418 – 440. · Zbl 0185.09401 · doi:10.1016/0021-8693(69)90040-4 · doi.org  Lawrence Levy, Torsion-free and divisible modules over non-integral-domains, Canad. J. Math. 15 (1963), 132 – 151. · Zbl 0108.04001 · doi:10.4153/CJM-1963-016-1 · doi.org  Lance W. Small, Orders in Artinian rings. II, J. Algebra 9 (1968), 266 – 273. · Zbl 0164.03904 · doi:10.1016/0021-8693(68)90025-2 · doi.org  Julius Martin Zelmanowitz, Endomorphism rings of torsionless modules, J. Algebra 5 (1967), 325 – 341. · Zbl 0155.36102 · doi:10.1016/0021-8693(67)90043-9 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.