MacWilliams' extension theorem for infinite rings
| Autor: | Friedrich Martin Schneider, Jens Zumbrägel |
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| Rok vydání: | 2017 |
| Předmět: |
FOS: Computer and information sciences
Pure mathematics Property (philosophy) Mathematics::Commutative Algebra Generalization Applied Mathematics General Mathematics Computer Science - Information Theory Information Theory (cs.IT) Mathematics::Rings and Algebras Artinian ring Extension (predicate logic) Mathematics - Rings and Algebras Pontryagin's minimum principle Character (mathematics) 16L60 16P20 94B05 Rings and Algebras (math.RA) FOS: Mathematics Finitary Quotient Mathematics |
| DOI: | 10.48550/arxiv.1709.06070 |
| Popis: | Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present note we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudo-injective and its finitary left socle embeds into the semisimple quotient. Providing a topological perspective on the MacWilliams property, we also show that the finitary left socle of a left Artinian ring embeds into the semisimple quotient if and only if it admits a finitarily left torsion-free character, if and only if the Pontryagin dual of the regular left module is almost monothetic. In conclusion, an Artinian ring has the MacWilliams property if and only if it is finitarily Frobenius, i.e., it is quasi-Frobenius and its finitary socle embeds into the semisimple quotient. Comment: 14 pages. To appear in Proceedings of the AMS |
| Databáze: | OpenAIRE |
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