Abstract
It is proved that A is a right distributive ring if and only if all quasiinjective right A-modules are Bezout left modules over their endomorphism rings if and only if for any quasiinjective right A-module M which is a Bezout left End (M)-module, every direct summand N of M is a Bezout left End(N)-module. If A is a right or left perfect ring, then all right A-modules are Bezout left modules over their endomorphism rings if and only if all right A-modules are distributive left modules over their endomorphism rings if and only if A is a distributive ring.
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REFERENCES
A. A. Tuganbaev, Semidistributive Modules and Rings, Kluwer Academic Publishers, Dordrecht-Boston-London, 1998.
C. Faith, Algebra: Rings, Modules, and Categories, vol. II, Springer-Verlag, Berlin, 1976.
V. Camillo, “Distributive modules,” J. Algebra, 36 (1975), no. 1, 16–25.
R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach Science Publishers, Philadelphia, 1991.
W. Stephenson, “Modules whose lattice of submodules is distributive,” Proc. London Math. Soc., 28 (1974), no. 2, 291–310.
L. Fuchs, Infinite Abelian Groups, vol. II, Academic Press, New York, 1973.
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Tuganbaev, A.A. Modules over Endomorphism Rings. Mathematical Notes 75, 836–847 (2004). https://doi.org/10.1023/B:MATN.0000030992.89821.2d
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DOI: https://doi.org/10.1023/B:MATN.0000030992.89821.2d