A Construction of Right Artinian Right Serial Rings

Abstract

A ring R is said to be right serial, if it is a direct sum of right ideals which are uniserial. A ring that is right serial need not be left serial. Right artinian, right serial ring naturally arise in the study of artinian rings satisfying certain conditions. For example, if an artinian ring R is such that all finitely generated indecomposable right R-modules are uniform or all finitely generated indecomposable left R-modules are local, then R is right serial. Such rings have been studied by many authors including Ivanov, Singh and Bleehed, and Tachikawa. In this paper, a universal construction of a class of indecomposable, non-local, basic, right artinian, right serial rings is given. The construction depends on a right artinian, right serial ring generating system X, which gives rise to a tensor ring T(L). It is proved that any basic right artinian, right serial ring is a homomorphic image of one such T(L).