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On some questions related to Koethe's nil ideal problem

Part of: Radicals and radical properties of rings

Published online by Cambridge University Press:  13 February 2015

M. A. Chebotar ,
P.-H. Lee  and
E. R. Puczyłowski
M. A. Chebotar
Affiliation:
Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242, USA (chebotar@math.kent.edu)
P.-H. Lee
Affiliation:
Department of Mathematics, National Taiwan University and , National Center for Theoretical Sciences, Taipei Office, Taipei 10617, Taiwan (phlee@math.ntu.edu.tw)
E. R. Puczyłowski
Affiliation:
Institute of Mathematics, University of Warsaw, 02-097 Warsaw, Banacha 2, Poland (edmundp@mimuw.edu.pl)
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Abstract

We study properties of two-sided and one-sided ideals of A-rings, i.e. rings that are sums of their nil left ideals. We show that the question as to whether one-sided ideals of A-rings are again A-rings is equivalent to the famous Koethe problem. We also obtain some results on another related open problem that asks whether annihilators of elements of non-zero A-rings are non-zero.

Keywords

Koethe problemnil ringsideals

MSC classification

Secondary: 16N40: Nil and nilpotent radicals, sets, ideals, rings 16N80: General radicals and rings
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 1 , February 2016 , pp. 57 - 64
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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References

1. Andrunakievich, V. A., The Dniester notebook: unsolved problems in the theory of rings and modules, Problems 5 and 6 (Akadamii Nauk Moldavskoi SSR, Kishinev, 1969) (in Russian).Google Scholar
2. Chebotar, M. A., Lee, P.-H. and Puczyłowski, E. R., On Andrunakievich's chain and Koethe's problem, Israel J. Math. 180 (2010), 119–128.CrossRefGoogle Scholar
3. Chebotar, M. A., Ke, W.-F., Lee, P.-H. and Puczyłowski, E. R., A linear algebra approach to Koethe’s problem and related questions, Linear Multilinear Alg. (2012), doi: 10.1080/03081087.2012.701298.Google Scholar
4. Ferrero, M. and Puczyłowski, E. R., The singular ideal and radicals, J. Austral. Math. Soc. A64 (1998), 195–209.Google Scholar
5. Gardner, B. J. and Wiegant, R., Radical theory of rings, Monographs and Textbooks in Pure and Applied Mathematics, Volume 261 (Dekker, New York, 2004).Google Scholar
6. Köthe, G., Die Struktur der Ringe, deren Restklassenring nach dem Radikal vollständig reduzibel ist, Math. Z. 32 (1930), 161–186.Google Scholar
7. Krempa, J., Logical connections between some open problems concerning nil rings, Fund. Math. 76 (1972), 121–130.Google Scholar
8. Puczyłowski, E. R., Questions related to Koethe’s nil ideal problem, in Algebra and its applications, Volume 419, pp. 269–283, Contemporary Mathematics (American Mathematical Society, Providence, RI, 2006).CrossRefGoogle Scholar
9. Rowen, L. H., Koethe's conjecture, in Ring Theory 1989, Proc. Ring Theory Symposium, Bar-Ilan University, Ramat Gan and the Division Algebra Workshop, Hebrew University, Jerusalem, 1988/1989, Israel Mathematical Conference Proceedings, Volume 1, pp. 193–202 (Weizmann, Jerusalem, 1989).Google Scholar