• Communications of the Korean Mathematical Society
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• v.15 no.3
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• pp.453-467
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• 2000

For any S-flat module RM(which will be called endoflat) with a commutaitve 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 follow; Rad(S) = { f$\in$S｜Imf=Mf is small in M} = {f$\in$S｜Imf $\leq$Rad(M)}. Additionally for any 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 quasi-injective endo-flat module has been studied too. Also some equivalent conditions for the semi-primitivity of any faithful endo-flat module RM with the open Jacobson Radical Rad(M) and those for the semi-simplicity of any faithful endo-flat quasi-injective module RM with the closed Socle Soc(M) have been studied.

• Communications of the Korean Mathematical Society
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• v.14 no.1
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• pp.1-12
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• 1999

The faithful bi-module \ulcornerM\ulcorner with its endomorphism ring End\ulcorner(M) such that M\ulcorner is flat (in other words, End\ulcorner(M)-flat, or endo-flat)and with a commutative ring R containing an identity has been studied in this paper. The chain conditions of a faithful endo-flat module \ulcornerM relative to those of the endomorphism ring End\ulcorner(M) having the zero annihilator of each non-zero endomorphism are studied.