Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1995-1273501-7
MathSciNet review: 1273501
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] F. W. Anderson and K. R. Fuller, Rings and categories of modules, Springer-Verlag, 1974. MR 0417223 (54:5281)
  • [2] E. P. Armendariz and J. K. Park, Self-injective rings with restricted chain conditions, Arch. Math. (Basel) 58 (1992), 24-33. MR 1139382 (92m:16002)
  • [3] J. Clark and D. V. Huynh, When is a semiperfect self-injective ring quasi-Frobenius?, J. Algebra 164 (1994), 531-542. MR 1275918 (95d:16006)
  • [4] N. V. Dung and P. F. Smith, $ \Sigma $-CS modules, Comm. Algebra 22 (1994), 83-93. MR 1255671 (94m:16003)
  • [5] C. Faith, Rings with ascending chain condition on annihilators, Nagoya Math. J. 27 (1966), 179-191. MR 0193107 (33:1328)
  • [6] -, Algebra II: Ring theory, Springer-Verlag, 1976. MR 0427349 (55:383)
  • [7] K. R. Goodearl and R. B. Warfield, An introduction to noncommutative Noetherian rings, London Math. Soc. Stud. Texts, no. 16, Cambridge Univ. Press, Cambridge, 1989. MR 1020298 (91c:16001)
  • [8] D. Jonah, Rings with minimum condition for principal right ideals have maximum condition for principal left ideals, Math. Z. 113 (1970), 106-112. MR 0260779 (41:5402)
  • [9] F. Kasch, Moduln and Ringe, Teubner Stuttgart, 1977. MR 0429963 (55:2971)
  • [10] T. Kato, Self-injective rings, Tôhoku Math. J. 19 (1967), 485-495. MR 0224648 (37:247)
  • [11] S. H. Mohamed and B. J. Müller, Continuous and discrete modules, London Math. Soc. Lecture Note Ser., no. 147, Cambridge Univ. Press, Cambridge, 1990. MR 1084376 (92b:16009)
  • [12] K. Oshiro, Lifting modules, extending modules and their application to QF-rings, Hokkaido Math. J. 13 (1984), 310-338. MR 764267 (86b:16008a)
  • [13] -, On Harada rings. I, Math J. Okayama Univ. 31 (1989), 161-178. MR 1043359 (91f:16025)
  • [14] B. L. Osofsky, A generalization of quasi-Frobenius rings, J. Algebra 4 (1966), 373-387. MR 0204463 (34:4305)
  • [15] Phan Dan, Right perfect rings with extending property on finitely generated free modules, Osaka J. Math. 26 (1989), 265-273. MR 1017585 (90k:16028)
  • [16] H. Tachikawa, Quasi-Frobenius rings and generalizations, QF-$ 3$ and QF-$ 1$ rings, Lecture Notes in Math., vol. 351, Springer, 1973. MR 0349740 (50:2233)
  • [17] R. Wisbauer, Foundations in module and ring theory, Gordon and Breach, Reading, 1991. MR 1144522 (92i:16001)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16L30, 16L60, 16P70

Retrieve articles in all journals with MSC: 16L30, 16L60, 16P70


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1273501-7
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society