The Jacobson Radical of the Endomorphism Ring, the Jacobson Radical, and the Socle of an Endo-Flat Module
Commun. Korean Math. Soc. 2000 Vol. 15, No. 3, 453-467
Soon-Sook Bae Kyungnam University
Abstract : For any $S-${\it flat} module ${}_RM$(which will be called {\it endo-flat}) with a commutative ring $R$ with identity, where $S$ is the endomorphism ring ${}_RM$, the fact that every epimorphism is an automorphism has been proved and the Jacobson Radical $Rad(S)$ of $S$ is described as follows; $ Rad(S) = \{ \; f \in S \; | \; Imf= Mf \; \text{is {\it small} in} \; M \; \} = \{ \; f \in S \; | \; Imf \leq Rad(M) \; \} . $ Additionally for any {\it quasi-injective endo-flat} module ${}_RM$, the fact that every monomorphism is an automorphism has been proved and the Jacobson Radical $Rad(S)$ for any {\it quasi-injective endo-flat} module has been studied too. Also some equivalent conditions for the semi-primitivity of any {\it faithful endo-flat} module ${}_RM$ with the {\it open} Jacobson Radical $Rad(M)$ and those for the semi-simplicity of of any {\it faithful endo-flat quasi-injective} module ${}_RM$ with the {\it closed} Socle $Soc(M)$ have been studied.
Keywords : open, closed submodule, endo-flat, quasi-injective, small (superfluous), large(essential), radical, socle, semi-primitive, semi-simple, and subdirect product
