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 i2-module is uniserial. When does this imply that the indecomposable injective left fl-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

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... ach primitive idempotent e, eRe has indecomposable 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.