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

Author:
Dinh Van Huynh

Journal:
Trans. Amer. Math. Soc. **347** (1995), 3131-3139

MSC:
Primary 16L30; Secondary 16L60, 16P70

DOI:
https://doi.org/10.1090/S0002-9947-1995-1273501-7

MathSciNet review:
1273501

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Abstract: A module is called a CS module if every submodule of is essential in a direct summand of . A ring is said to be right (countably) -CS if any direct sum of (countably many) copies of the right -module is CS. It is shown that for a right countably -CS ring the following are equivalent: (i) is right -CS, (ii) has ACC or DCC on projective principal right ideals, (iii) has finite right uniform dimension and ACC or DCC holds on projective uniform principal right ideals of , (iv) is semiperfect. From results of Oshiro [12], [13], under these conditions, is left artinian and QF-. As a consequence, a ring is quasi-Frobenius if it is right countably -CS, semiperfect and no nonzero projective right ideals are contained in its Jacobson radical.

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DOI:
https://doi.org/10.1090/S0002-9947-1995-1273501-7

Article copyright:
© Copyright 1995
American Mathematical Society