1. ENDOMORPHISM RING OF A UNISERIAL MODULE Irawati Algebra Research Group Faculty of Mathematics and Natural Sciences, ITB, Bandung, Indonesia

2. Uniserial Module Uniserial Module : a module with unique composition series Uniserial module is indecomposable

3. Theorem 1 Let M be a uniserial module. Then there are only two possibilities for the elements of End(M), nilpotent or isomorphism

4. .. So f is nilpotent.. with and with. If there is an r such that, write Let. Because of, then

5. Theorem 2. If S is a ring with only two types of elements, nilpotent and invertible, then S is a local ring.

6. References Anderson, F. W., and K. R. Fuller, 1991, Rings and Categories of Modules, Springer Verlag Passman, D.S., 1991, A Course in Ring Theory, Worth and Brooks/Cole. Irawati, Matrices over Uniserial Ring, International Mathematical Forum, 3, 2008, no. 29, 1401 - 1407