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Modules over the endomorphism ring of a finitely generated projective module


Author: F. L. Sandomierski
Journal: Proc. Amer. Math. Soc. 31 (1972), 27-31
DOI: https://doi.org/10.1090/S0002-9939-1972-0288137-4
MathSciNet review: 0288137
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Abstract | References | Additional Information

Abstract: Let $ {P_R}$ be a projective module with trace ideal $ T$. An $ R$-module $ {X_R}$ is $ T$-accessible if $ XT = X.{\text{ If }}{P_R}$ is finitely generated projective and $ C$ is the $ R$-endomorphism ring of $ {P_R}$, such that $ _C{P_R}$, then for $ {X_R}$, Horn $ {({P_R},{X_R})_C}$ is artinian (noetherian) if and only if $ {X_R}$ satisfies the minimum (maximum) condition on $ T$-accessible submodules. Further, if $ {X_R}$ is $ T$-accessible then Hom $ {({P_R},{X_R})_C}$ is finitely generated if and only if $ {X_R}$ is finitely generated.


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

  • [1] H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466-188. MR 28 #1212. MR 0157984 (28:1212)
  • [2] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N.J., 1956. MR 17, 1040. MR 0077480 (17:1040e)
  • [3] A. W. Goldie, The structure of prime rings under ascending chain conditions, Proc. London Math. Soc. (3) 8 (1958), 589-608. MR 21 #1988. MR 0103206 (21:1988)
  • [4] L. Silver, Noncommutative localizations and applications, J. Algebra 7 (1967) 44-76. MR 36 #205. MR 0217114 (36:205)
  • [5] K. Morita, Adjoint pairs of functors and Frobenius extensions, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 9 (1965), 40-71. MR 32 #7597. MR 0190183 (32:7597)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0288137-4
Keywords: Projective module, chain conditions
Article copyright: © Copyright 1972 American Mathematical Society

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