Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Finiteness conditions for projective and injective modules


Author: Joe W. Fisher
Journal: Proc. Amer. Math. Soc. 40 (1973), 389-394
MSC: Primary 16A50
DOI: https://doi.org/10.1090/S0002-9939-1973-0335578-3
MathSciNet review: 0335578
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Does Hopkins' theorem extend to projective modules, i.e., are projective Artinian modules Noetherian? An example is given to answer this question in the negative; however, we show that the answer is affirmative for certain large classes of projective modules. Dually, are injective Noetherian modules Artinian? Again the answer is negative; nevertheless, we provide an affirmative answer for certain classes of injective modules.


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

  • [1] S. A. Amitsur, Prime rings having polynomial identities with arbitrary coefficients, Proc. London. Math. Soc. (3) 17 (1967), 470-486. MR 36 #209. MR 0217118 (36:209)
  • [2] H. Bass, Finitistic dimension and a homological generalization of semiprimary rings, Trans. Amer. Math. Soc. 95 (1960), 466-488. MR 28 #1212. MR 0157984 (28:1212)
  • [3] -, Algebraic $ K$-theory, Benjamin, New York, 1968. MR 40 #2736.
  • [4] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N.J., 1956. MR 17, 1040. MR 0077480 (17:1040e)
  • [5] C. Faith and E. A. Walker, Direct-sum representations of injective modules, J. Algebra 5 (1967), 203-221. MR 34 #7575. MR 0207760 (34:7575)
  • [6] J. W. Fisher, Decomposition theories for modules, Trans. Amer. Math. Soc. 145 (1969), 241-269. MR 40 #5656. MR 0252436 (40:5656)
  • [7] -, Endomorphism rings of modules, Notices Amer. Math. Soc. 18 (1971), 619-620. Abstract #71T-A85.
  • [8] -, Nil subrings of endomorphism rings of modules, Proc. Amer. Math. Soc. 34 (1972), 75-78. MR 45 #1960. MR 0292878 (45:1960)
  • [9] R. Gordon, Ring theory, Academic Press, New York, 1972. MR 0330129 (48:8467)
  • [10] M. Harada, On semi-simple abelian categories, Osaka J. Math. 7 (1970), 89-95. MR 42 #7748. MR 0272867 (42:7748)
  • [11] I. Kaplansky, Fields and rings, Univ. of Chicago Press, Chicago, Ill., 1969. MR 42 #4345. MR 0269449 (42:4345)
  • [12] -, Commutative rings, Allyn and Bacon, Boston, Mass., 1970. MR 40 #7234. MR 0254021 (40:7234)
  • [13] F. Kasch and E. A. Mares, Eine Kennzeichnung semi-perfekter Moduln, Nagoya Math. J. 27 (1966), 525-529. MR 33 #7376. MR 0199227 (33:7376)
  • [14] R. Miller and D. Turnidge, Some examples from infinite matrix rings, Proc. Amer. Math. Soc. 38 (1973), 65-67. MR 0310001 (46:9104)
  • [15] Y. Miyashita, Quasi-projective modules, perfect modules, and a theorem for modular lattices, J. Fac. Sci. Hokkaido Univ. Ser. I 19 (1966), 86-110. MR 35 #4254. MR 0213390 (35:4254)
  • [16] D. G. Northcott, Lessons on rings, modules, and multiplicities, Cambridge Univ. Press, London, 1968. MR 38 #144. MR 0231816 (38:144)
  • [17] E. C. Posner, Prime rings satisfying a polynomial identity, Proc. Amer. Math. Soc. 11 (1960), 180-183. MR 22 #2626. MR 0111765 (22:2626)
  • [18] F. L. Sandomierski, Modules over the endomorphism ring over a finitely generated projective module, Proc. Amer. Math. Soc. 31 (1972), 27-31. MR 44 #5335. MR 0288137 (44:5335)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 16A50

Retrieve articles in all journals with MSC: 16A50


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0335578-3
Keywords: Projective, injective, Artinian, Noetherian, endomorphism rings of modules, semiprimary, P.I.-rings
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society