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Nonsingular retractable modules and their endomorphism rings

  • Soumaya Makdissi Khuri (a1)
Abstract

A module RM is said to be retractable if HomR (M, U) ≠ 0 for each nonzero submodule U of M. M is said to be a CS module if every complement submodule of M is a direct summand in M. Retractable modules are compared to nondegenerate modules on the one hand and to e–retractable modules on the other (nondegenerate implies retractable implies e–retractable); and it is shown that if M is nonsingular and retractable, then EndRM is a left CS ring if and only if M is a CS module.

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[1]Albu, T. and Nastasescu, C., Relative finiteness in module theory (Dekker, New York, 1984).
[2]Chatters, A.W. and Khuri, S.M., ‘Endomorphism rings of modules over nonsingular CS rings’, J. London Math. Soc. 21 (1980), 434444.
[3]Faith, C., Lectures on injective modules and quotient rings 246, Lecture Notes in Mathematics (Springer-Verlag, Heidelberg, Berlin, New York, 1972).
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[5]Khuri, S.M., ‘Endomorphism rings of nonsingular modules’, Ann. Sci. Math. Quebec 4 (1980), 145152.
[6]Khuri, S.M., ‘Correspondence theorems for modules and their endomorphism rings’, J. Algebra 122 (1989), 380396.
[7]Utumi, Y., ‘On rings of which any one-sided quotient ring is two-sided’, Proc. Amer. Math. Soc. 14 (1963), 141147.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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