Rings Whose Modules Are (D₁₂)
by
Derya Keskin
Hacettepe University, Department of Mathematics, 06532 Beytepe, Ankara, Turkey

In this work we assume that R is an assosiative ring with identity and all R-modules are unitary right R-modules.

Let M be an R-module. M is is called \oplus-supplemented if for every submodule N of M there is a direct summand K of M such that M=N+K and N \cap K is small in K [MM]. As a proper generalization of \oplus-supplemented modules Keskin and Xue [KX] call an R-module M (D₁₂) if, for every submodule N of M, there exists a direct summand K of M and an epimorphism \alpha: M/K --> M/N such that Ker\alpha is small in M/K. Keskin, Smith and Xue proved in [KSX] that a ring R is serial if and only if every finitely presented right R-module and finitely presented left R-module is \oplus-supplemented, and R is artinian serial if and only if every right R-module and left R-module is \oplus-supplemented. In this work we prove that the \oplus-supplemented condition can be replaced by (D₁₂) condition.

References

[MM] S.H. Mohamed and B.J. Müller, Continuous and discrete modules, London Math. Soc. Lecture Note Series 147, Cambridge Univ. Press, Cambridge (1990).

[KX

Date received: July 26, 2000