
Vietnam Journal of Mathematics 36:2(2008)
229238

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 nonsmall right Rmodule
contains a nonzero 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 QF3 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,
cosmall module.


Established
by Vietnam Academy of Science and Technology & Vietnam Mathematical
Society
Published
by Springer since January 2013

