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Baer and Quasi-Baer Modules over Some Classes of Rings

Abdelfattah Haily|Hamid Rahnaoui
경북대학교 자연과학대학 수학과|2011.12|Kyungpook Mathematical Journal|제51권 제4호|pp. 375-384

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키워드

IdempotentAnnihilatorEndomorphismBaer moduleK-nonsingular module
주제분류수학

초록

We study Baer and quasi-Baer modules over some classes of rings. We also introduce a new class of modules called AI-modules, in which the kernel of every nonzero endomorphism is contained in a proper direct summand. The main results obtained here are: (1) A module is Baer iff it is an AI-module and has SSIP. (2) For a perfect ring R, the direct sum of Baer modules is Baer iff R is primary decomposable. (3) Every injective R-module is quasi-Baer iff R is a QI-ring.

Abstract

We study Baer and quasi-Baer modules over some classes of rings. We also introduce a new class of modules called AI-modules, in which the kernel of every nonzero endomorphism is contained in a proper direct summand. The main results obtained here are: (1) A module is Baer iff it is an AI-module and has SSIP. (2) For a perfect ring R, the direct sum of Baer modules is Baer iff R is primary decomposable. (3) Every injective R-module is quasi-Baer iff R is a QI-ring.

참고문헌 (14)

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  8. I. Kaplansky, "Rings of operators" (1968), Benjamin
  9. S. T. Rizvi, "Baer and quasi-Baer modules", Comm. Algebra, 32, pp.103~123 (2004)
  10. S. T. Rizvi, "On K-nonsingular modules and applications", Comm. Alge-bra, 35, pp.2960~2982 (2007)
  11. S. T. Rizvi, "On direct sums of Baer modules", J. Algebra, 321, pp.628~696 (2009)
  12. T. Y. Lam, "Lectures on modules and rings, In Graduate Texts in Mathematics, 189" (1999), Springer-Verlag
  13. G. V. Wilson, "Modules with the summand intersection property", Comm. Alg, 14(1), pp.21~38 (1986)
  14. R. Wisbauer, "Foundations of module and ring theory" (1991), Gordon and Beach Science Publishers

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