J Austral Math Soc Ser A 44 pp242--251, 1988.
(Received 31 October 1985; revised 5 January 1987)
A module is uniserial if its lattice of submodules is linearly ordered, and a ring R is left serial if R is a direct sum of uniserial left ideals. The following problem is considered. Suppose the injective hull of each simple left R-module is uniserial . When does this imply that the indecomposable injective left R-modules are uniserial? An affirmative answer is known when R is commutative and when R is Artinian. The following result is proved. Let R be a left serial ring and suppose that for each primitive idempotent e, eRe has idemposable injective left modules uniserial. The following conditions are equivalent. (a) The injective hull of each simple left R-module is uniserial. (b) Every indecomposable injective left R-module is uniserial. (c) Every finitely generated left R-module is serial. The rest of the paper is devoted to a study of some non-Artinian serial rings which serve to illustrate this theorem.
1980 AMS Subject Classification: primary 16A52, 16A53; secondary 16A04
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