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Vietnam Journal of Mathematics 36:2(2008) 125-136 T1 Theorems for Inhomogeneous Besov and Triebel-Lizorkin Spaces over Space of Homogeneous Type   Yanchang Han 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: 42B25, 42B35, 46E35. Keywords: $T1$ theorem, inhomogeneous Besov and Triebel-Lizorkin spaces, spaces of homogeneous type.

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