Indecomposable decompositions and the minimal direct summand containing the nilpotents

Abstract

It is well known that an indecomposable right ideal decomposition of a ring is not necessarily unique. In this paper we show that the reduced right ideals of such a decomposition are unique up to isomorphism and the remainder of the decomposition forms the unique MDSN. In the main theorem we use triangular matrices to prove that a ring with an indecomposable decomposition is basically composed of a nilpotent ring, a ring (containing a unity) with an indecomposable decomposition which equals its MDSN, and a direct sum of indecomposable reduced rings with unity.