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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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