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Vietnam Journal of Mathematics 36:2(2008) 229-238  On Harada Rings and Serial Artinian Rings Thanakarn Soonthornkrachang, Phan Dan, Nguyen Van Sanh, and Kar Ping Shum 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.   2000 Mathematics Subject Classification: 16D50, 16D70, 16D80. Keywords: Harada ring, Artinian rign, small module, co-small module.

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