Geodesic
Rings whose Indecomposable Injective Modules are Uniserial
Canadian journal of mathematics, Tome 34 (1982) no. 4, pp. 797-805

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A module is uniserial in case its submodules are linearly ordered by inclusion. A ring R is left (right) serial if it is a direct sum of uniserial left (right) R-modules. A ring R is serial if it is both left and right serial. It is well known that for artinian rings the property of being serial is equivalent to the finitely generated modules being a direct sum of uniserial modules [8]. Results along this line have been generalized to more arbitrary rings [6], [13].This article is concerned with investigating rings whose indecomposable injective modules are uniserial. The following question is considered which was first posed in [4]. If an artinian ring R has all indecomposable injective modules uniserial, does this imply that R is serial? The answer is yes if R is a finite dimensional algebra over a field. In this paper it is shown, provided R modulo its radical is commutative, that R has every left indecomposable injective uniserial implies that R is right serial. 
Rings whose Indecomposable Injective Modules are Uniserial. Canadian journal of mathematics, Tome 34 (1982) no. 4, pp. 797-805. doi: 10.4153/CJM-1982-055-3
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