MacWilliams’ extension theorem for infinite rings
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- by Friedrich Martin Schneider and Jens Zumbrägel PDF
- Proc. Amer. Math. Soc. 147 (2019), 947-961 Request permission
Abstract:
Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present paper 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 pseudoinjective 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.References
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Additional Information
- Friedrich Martin Schneider
- Affiliation: Institute of Algebra, TU Dresden, 01062 Dresden, Germany
- MR Author ID: 963016
- Email: martin.schneider@tu-dresden.de
- Jens Zumbrägel
- Affiliation: Faculty of Computer Science and Mathematics, University of Passau, 94032 Passau, Germany
- MR Author ID: 843678
- Email: jens.zumbraegel@uni-passau.de
- Received by editor(s): September 16, 2017
- Received by editor(s) in revised form: September 18, 2017, April 2, 2018, and May 30, 2018
- Published electronically: November 16, 2018
- Additional Notes: The first author acknowledges funding of the Excellence Initiative by the German Federal and State Governments.
- Communicated by: Jerzy Weyman
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 947-961
- MSC (2010): Primary 16L60, 16P20, 94B05
- DOI: https://doi.org/10.1090/proc/14343
- MathSciNet review: 3896045