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OPENLY SEMIPRIMITIVE PROJECTIVE MODULE

Bae, Soon-Sook

  • Published : 2004.10.01

Abstract

In this paper, a left module over an associative ring with identity is defined to be openly semiprimitive (strongly semiprimitive, respectively) by the zero intersection of all maximal open fully invariant submodules (all maximal open submodules which are fully invariant, respectively) of it. For any projective module, the openly semiprimitivity of the projective module is an equivalent condition of the semiprimitivity of endomorphism ring of the projective module and the strongly semiprimitivity of the projective module is an equivalent condition of the endomorphism ring of the projective module being a sub direct product of a set of subdivisions of division rings.

Keywords

free;projective;semiprimitive;openly (strongly) semi-primitive module;maximal open submodule;prime endomorphism

References

  1. H. Bass, Finistic Dimension and a Homological Generalization of Semiprimary Rings, Trans. Amer. Math. Soc. 95 (1960), 466-488. https://doi.org/10.1090/S0002-9947-1960-0157984-8
  2. I. N. Herstein, Noncommutative rings, MAA Notes, 1968.
  3. R. Ware and J. Zelmanowitz, The Jacobson Radical of the Endomorphism Ring of a Projective Module, Proc. Amer. Math. Soc. 26 (1970), 15-20. https://doi.org/10.1090/S0002-9939-1970-0262281-8
  4. R. Ware, Endomorphism Rings of Projective Modules, Trans. Amer. Math. Soc. 55 (1971), 233-256. https://doi.org/10.1090/S0002-9947-1971-0274511-2
  5. F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, New York; Spring-Verlag, 1973.
  6. I. Beck, On Modules whose Endomorphism ring is Local, Israel J. Math. 29 (1978), no. 4, 393-407. https://doi.org/10.1007/BF02761177
  7. T. W. Hungerford, Algebra, New York; Spring -Verlag, 1984.
  8. S.-S. Bae, On Reject of Subdirect Product, Kyungnam University, Thesis Collection 14 (1987), 7-10.
  9. S.-S. Bae, Certain Discriminations of Prime Endomorphism and Prime Matrix, East Asian Math. J. 14 (1998 no. 2), 259-268.
  10. S.-S. Bae, Generalized Schur's Lemmas, Journal of the Graduate School, Kyungnam University 16 (2001 no. 2), 7-11.

Cited by

  1. REGULAR ENDOMORPHISM RINGS OF PROJECTIVE MODULES vol.30, pp.4, 2008, https://doi.org/10.5831/HMJ.2008.30.4.617