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ON COMMUTATIVE REDUCED FILIAL RINGS

Part of: Modules, bimodules and ideals

Published online by Cambridge University Press:  21 October 2009

R. R. ANDRUSZKIEWICZ  and
K. PRYSZCZEPKO
R. R. ANDRUSZKIEWICZ
Affiliation:
Institute of Mathematics, University of Białystok, 15-267 Białystok, Akademicka 2, Poland (email: randrusz@math.uwb.edu.pl)
K. PRYSZCZEPKO*
Affiliation:
Institute of Mathematics, University of Białystok, 15-267 Białystok, Akademicka 2, Poland (email: karolp@math.uwb.edu.pl)
*
✉For correspondence; e-mail: karolp@math.uwb.edu.pl
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Abstract

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A ring in which every accessible subring is an ideal is called filial. We continue the study of commutative reduced filial rings started in [R. R. Andruszkiewicz and K. Pryszczepko, ‘A classification of commutative reduced filial rings’, Comm. Algebra to appear]. In particular we describe the Noetherian commutative reduced rings and construct nontrivial examples of commutative reduced filial rings without ideals which are domains.

Keywords

idealfilial ringreduced ringp-adic numbers

MSC classification

Secondary: 16D25: Ideals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 2 , April 2010 , pp. 310 - 316
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

References

[1]Andruszkiewicz, R. R., ‘The classification of integral domains in which the relation of being an ideal is transitive’, Comm. Algebra 31 (2003), 2067–2093.CrossRefGoogle Scholar
[2]Andruszkiewicz, R. R. and Pryszczepko, K., ‘A classification of commutative reduced filial rings’, Comm. Algebra to appear.Google Scholar
[3]Andruszkiewicz, R. R. and Puczyłowski, E. R., ‘On filial rings’, Portugal. Math. 45 (1988), 139–149.Google Scholar
[4]Andruszkiewicz, R. R. and Sobolewska, M., ‘Commutative reduced filial rings’, Algebra Discrete Math. 3 (2007), 18–26.Google Scholar
[5]Ehrlich, G., ‘Filial rings’, Portugal. Math. 42 (1983/1984), 185–194.Google Scholar
[6]Filipowicz, M. and Puczyłowski, E. R., ‘Left filial rings’, Algebra Colloq. 11 (2004), 335–344.Google Scholar
[7]Filipowicz, M and Puczyłowski, E. R., ‘On filial and left filial rings’, Publ. Math. Debrecen. 66 (2005), 257–267.CrossRefGoogle Scholar
[8]Gardner, B. J. and Wiegandt, R., Radical Theory of Rings (Marcel Dekker, New York, 2004).Google Scholar
[9]Halmos, P., Lectures on Boolean Algebras (Van Nostrand, Princeton, NJ, 1963).Google Scholar
[10]Sands, A. D., ‘On ideals in over-rings’, Publ. Math. Debrecen. 35 (1988), 273–279.CrossRefGoogle Scholar
[11]Satyanarayana, M., ‘Rings with primary ideals as maximal ideals’, Math. Scand. 20 (1967), 52–54.CrossRefGoogle Scholar
[12]Szász, F. A., Radical of Rings (Akadémiai Kiadó, Budapest, 1981).Google Scholar
[13]Veldsman, S., ‘Extensions and ideals of rings’, Publ. Math. Debrecen. 38 (1991), 297–309.CrossRefGoogle Scholar