The Jacobson Radical of the Endomorphism Ring, the Jacobson Radical, and the Socle of an Endo-Flat Module

摘要

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 ; ext{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.