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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A right countably sigma-CS ring with ACC or DCC on projective principal right ideals is left Artinian and QF-$ 3$


Author: Dinh Van Huynh
Journal: Trans. Amer. Math. Soc. 347 (1995), 3131-3139
MSC: Primary 16L30; Secondary 16L60, 16P70
MathSciNet review: 1273501
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Abstract: A module $ M$ is called a CS module if every submodule of $ M$ is essential in a direct summand of $ M$. A ring $ R$ is said to be right (countably) $ \Sigma $-CS if any direct sum of (countably many) copies of the right $ R$-module $ R$ is CS. It is shown that for a right countably $ \Sigma $-CS ring $ R$ the following are equivalent: (i) $ R$ is right $ \Sigma $-CS, (ii) $ R$ has ACC or DCC on projective principal right ideals, (iii) $ R$ has finite right uniform dimension and ACC or DCC holds on projective uniform principal right ideals of $ R$, (iv) $ R$ is semiperfect. From results of Oshiro [12], [13], under these conditions, $ R$ is left artinian and QF-$ 3$. As a consequence, a ring $ R$ is quasi-Frobenius if it is right countably $ \Sigma $-CS, semiperfect and no nonzero projective right ideals are contained in its Jacobson radical.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1995-1273501-7
PII: S 0002-9947(1995)1273501-7
Article copyright: © Copyright 1995 American Mathematical Society