On left Köthe rings and a generalization of a Köthe-Cohen-Kaplansky theorem
HTML articles powered by AMS MathViewer
- by M. Behboodi, A. Ghorbani, A. Moradzadeh-Dehkordi and S. H. Shojaee PDF
- Proc. Amer. Math. Soc. 142 (2014), 2625-2631 Request permission
Abstract:
In this paper, we obtain a partial solution to the following question of Köthe: For which rings $R$ is it true that every left (or both left and right) $R$-module is a direct sum of cyclic modules? Let $R$ be a ring in which all idempotents are central. We prove that if $R$ is a left Köthe ring (i.e., every left $R$-module is a direct sum of cyclic modules), then $R$ is an Artinian principal right ideal ring. Consequently, $R$ is a Köthe ring (i.e., each left and each right $R$-module is a direct sum of cyclic modules) if and only if $R$ is an Artinian principal ideal ring. This is a generalization of a Köthe-Cohen-Kaplansky theorem.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
- Stephen U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 (1960), 457–473. MR 120260, DOI 10.1090/S0002-9947-1960-0120260-3
- I. S. Cohen and I. Kaplansky, Rings for which every module is a direct sum of cyclic modules, Math. Z. 54 (1951), 97–101. MR 43073, DOI 10.1007/BF01179851
- Carl Faith, On Köthe rings, Math. Ann. 164 (1966), 207–212. MR 195903, DOI 10.1007/BF01360245
- Kent R. Fuller, On rings whose left modules are direct sums of finitely generated modules, Proc. Amer. Math. Soc. 54 (1976), 39–44. MR 393133, DOI 10.1090/S0002-9939-1976-0393133-6
- Phillip Griffith, On the decomposition of modules and generalized left uniserial rings, Math. Ann. 184 (1969/70), 300–308. MR 257136, DOI 10.1007/BF01350858
- Jebrel M. Habeb, A note on zero commutative and duo rings, Math. J. Okayama Univ. 32 (1990), 73–76. MR 1112012
- S. K. Jain and Ashish K. Srivastava, Rings characterized by their cyclic modules and right ideals: A survey-I. (see http://euler.slu.edu/~srivastava/articles.html)
- Gottfried Köthe, Verallgemeinerte Abelsche Gruppen mit hyperkomplexem Operatorenring, Math. Z. 39 (1935), no. 1, 31–44 (German). MR 1545487, DOI 10.1007/BF01201343
- 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
- Tadasi Nakayama, Note on uni-serial and generalized uni-serial rings, Proc. Imp. Acad. Tokyo 16 (1940), 285–289. MR 3618
- W. K. Nicholson and E. Sánchez Campos, Rings with the dual of the isomorphism theorem, J. Algebra 271 (2004), no. 1, 391–406. MR 2022487, DOI 10.1016/j.jalgebra.2002.10.001
- Lev Sabinin, Larissa Sbitneva, and Ivan Shestakov (eds.), Non-associative algebra and its applications, Lecture Notes in Pure and Applied Mathematics, vol. 246, Chapman & Hall/CRC, Boca Raton, FL, 2006. MR 2203689, DOI 10.1201/9781420003451
- A. A. Tuganbaev, Rings over which every module is a direct sum of distributive modules, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1 (1980), 61–64, 94 (Russian, with English summary). MR 563049
- Robert B. Warfield Jr., Rings whose modules have nice decompositions, Math. Z. 125 (1972), 187–192. MR 289487, DOI 10.1007/BF01110928
- Robert Wisbauer, Foundations of module and ring theory, Revised and translated from the 1988 German edition, Algebra, Logic and Applications, vol. 3, Gordon and Breach Science Publishers, Philadelphia, PA, 1991. A handbook for study and research. MR 1144522
- Birge Zimmermann-Huisgen, Rings whose right modules are direct sums of indecomposable modules, Proc. Amer. Math. Soc. 77 (1979), no. 2, 191–197. MR 542083, DOI 10.1090/S0002-9939-1979-0542083-3
Additional Information
- M. Behboodi
- Affiliation: Department of Mathematical Sciences, Isfahan University of Technology, P.O. Box 84156-83111, Isfahan, Iran – and – School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran
- Email: mbehbood@cc.iut.ac.ir
- A. Ghorbani
- Affiliation: Department of Mathematical Sciences, Isfahan University of Technology, P.O. Box 84156-83111, Isfahan, Iran
- Email: a_ghorbani@cc.iut.ac.ir
- A. Moradzadeh-Dehkordi
- Affiliation: Department of Mathematical Sciences, Isfahan University of Technology, P.O. Box 84156-83111, Isfahan, Iran
- Email: a.moradzadeh@math.iut.ac.ir
- S. H. Shojaee
- Affiliation: Department of Mathematical Sciences, Isfahan University of Technology, P.O. Box 84156-83111, Isfahan, Iran
- Email: hshojaee@math.iut.ac.ir
- Received by editor(s): June 15, 2010
- Received by editor(s) in revised form: March 6, 2011, March 28, 2011, and August 27, 2012
- Published electronically: April 22, 2014
- Additional Notes: The research of the first author was in part supported by a grant from IPM (No. 89160031).
The first author is the corresponding author - Communicated by: Birge Huisgen-Zimmermann
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 2625-2631
- MSC (2010): Primary 16D10, 16D70, 16P20; Secondary 16N60
- DOI: https://doi.org/10.1090/S0002-9939-2014-11158-0
- MathSciNet review: 3209318