MacWilliams’ extension theorem for infinite rings
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- by Friedrich Martin Schneider and Jens Zumbrägel
- Proc. Amer. Math. Soc. 147 (2019), 947-961
- DOI: https://doi.org/10.1090/proc/14343
- Published electronically: November 16, 2018
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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
- Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, 2nd ed., Graduate Texts in Mathematics, vol. 13, Springer-Verlag, New York, 1992. MR 1245487, DOI 10.1007/978-1-4612-4418-9
- Alexander Arhangel′skii and Mikhail Tkachenko, Topological groups and related structures, Atlantis Studies in Mathematics, vol. 1, Atlantis Press, Paris; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. MR 2433295, DOI 10.2991/978-94-91216-35-0
- H. Bass, $K$-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math. 22 (1964), 5–60. MR 174604
- D. van Dantzig, Zur topologischen Algebra, Math. Ann. 107 (1933), no. 1, 587–626 (German). MR 1512818, DOI 10.1007/BF01448911
- Anton Deitmar and Siegfried Echterhoff, Principles of harmonic analysis, Universitext, Springer, New York, 2009. MR 2457798
- Hai Quang Dinh and Sergio R. López-Permouth, On the equivalence of codes over finite rings, Appl. Algebra Engrg. Comm. Comput. 15 (2004), no. 1, 37–50. MR 2142429, DOI 10.1007/s00200-004-0149-5
- Hai Quang Dinh and Sergio R. López-Permouth, On the equivalence of codes over rings and modules, Finite Fields Appl. 10 (2004), no. 4, 615–625. MR 2094161, DOI 10.1016/j.ffa.2004.01.001
- Noyan Er, Surjeet Singh, and Ashish K. Srivastava, Rings and modules which are stable under automorphisms of their injective hulls, J. Algebra 379 (2013), 223–229. MR 3019253, DOI 10.1016/j.jalgebra.2013.01.021
- Christian Gottlieb, On finite unions of ideals and cosets, Comm. Algebra 22 (1994), no. 8, 3087–3097. MR 1272374, DOI 10.1080/00927879408825014
- Paul R. Halmos and H. Samelson, On monothetic groups, Proc. Nat. Acad. Sci. U.S.A. 28 (1942), 254–258. MR 6543, DOI 10.1073/pnas.28.6.254
- A. Roger Hammons Jr., P. Vijay Kumar, A. R. Calderbank, N. J. A. Sloane, and Patrick Solé, The $\textbf {Z}_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory 40 (1994), no. 2, 301–319. MR 1294046, DOI 10.1109/18.312154
- Manabu Harada, Self mini-injective rings, Osaka Math. J. 19 (1982), no. 3, 587–597. MR 676239
- Thomas Honold, Characterization of finite Frobenius rings, Arch. Math. (Basel) 76 (2001), no. 6, 406–415. MR 1831096, DOI 10.1007/PL00000451
- Miodrag Cristian Iovanov, Frobenius-Artin algebras and infinite linear codes, J. Pure Appl. Algebra 220 (2016), no. 2, 560–576. MR 3399378, DOI 10.1016/j.jpaa.2015.05.030
- T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999. MR 1653294, DOI 10.1007/978-1-4612-0525-8
- Florence Jessie MacWilliams, COMBINATORIAL PROBLEMS OF ELEMENTARY ABELIAN GROUPS, ProQuest LLC, Ann Arbor, MI, 1962. Thesis (Ph.D.)–Radcliffe College. MR 2939359
- Tadasi Nakayama, On Frobeniusean algebras. II, Ann. of Math. (2) 42 (1941), 1–21. MR 4237, DOI 10.2307/1968984
- W. K. Nicholson and M. F. Yousif, Mininjective rings, J. Algebra 187 (1997), no. 2, 548–578. MR 1430998, DOI 10.1006/jabr.1996.6796
- Donald S. Passman, Infinite group rings, Pure and Applied Mathematics, vol. 6, Marcel Dekker, Inc., New York, 1971. MR 0314951
- Jay A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121 (1999), no. 3, 555–575. MR 1738408
- Jay A. Wood, Code equivalence characterizes finite Frobenius rings, Proc. Amer. Math. Soc. 136 (2008), no. 2, 699–706. MR 2358511, DOI 10.1090/S0002-9939-07-09164-2
Bibliographic 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