Abstract
We say that \({\mathcal{A}}\) is a ring with duality for simple modules, or simply a DSM-ring, if, for every simple right (left) \({\mathcal{A}}\)-module U, the dual module U* is a simple left (right) \({\mathcal{A}}\)-module. We prove that a semiperfect ring is a DSM-ring if and only if it admits a Nakayama permutation. We introduce the notion of a monomial ideal of a semiperfect ring and study the structure of hereditary semiperfect rings with monomial ideals. We consider perfect rings with monomial socles.
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Dokuchaev, M.A., Kirichenko, V.V. Quasi-Frobenius Rings and Nakayama Permutations of Semiperfect Rings. Ukrainian Mathematical Journal 54, 1112–1125 (2002). https://doi.org/10.1023/A:1022062325089
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DOI: https://doi.org/10.1023/A:1022062325089