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Vietnam Journal of Mathematics 36:2(2008)
229-238
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On Harada Rings and Serial Artinian Rings
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Thanakarn Soonthornkrachang,
Phan Dan, Nguyen Van Sanh, and Kar Ping Shum
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Abstract. A ring R is called a right Harada ring if it is right Artinian and
every non-small right R-module
contains a non-zero injective submodule. The first result in our paper is
the following: Let R be a right
perfect ring. Then R is a right
Harada ring if and only if every cyclic module is a direct sum of an
injective module and a small module; if and only if every local module is
either injective or small. We also prove that a ring R is QF if and only if every cyclic module is a direct sum of a
projective injective module and a small module; if and only if every local
module is either projective injective or small. Finally, a right QF-3 right
perfect ring R is serial Artinian
if and only if every right ideal is a direct sum of a projective module and
a singular uniserial module.
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2000 Mathematics Subject Classification: 16D50, 16D70,
16D80.
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Keywords: Harada ring, Artinian rign, small module,
co-small module.
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Established
by Vietnam Academy of Science and Technology & Vietnam Mathematical
Society
Published
by Springer since January 2013
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