Abstract
A module over an artinian ring is uniserial if it has a unique composition series, and an artinian ring is serial if each of its indecomposable projective modules is uniserial. Fuller [4, Theorem 5.4] showed that an artinian ring R is serial if and only if each of left indecomposable projective and injective R-modules is uniserial. The following question was raised in 4, p.134: Is an artinian ring R necessarily serial if each of its indecomposable injective modules is uniserial? Example 1 in this note answers this question in the negative
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