Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T18:29:59.464Z Has data issue: false hasContentIssue false
  • English
  • Français

On semiprime segments of rings

Part of: Modules, bimodules and ideals Rings and algebras with additional structure

Published online by Cambridge University Press:  09 April 2009

R. Mazurek  and
G. Törner
R. Mazurek
Affiliation:
Faculty of Computer Science, Bialystok Technical University, Wiejska 45A, 15–351 Bialystok, Poland, e-mail: [email protected]
G. Törner
Affiliation:
Institut für Mathematik, Fakultät 4, Universität Duisburg-Essen, Campus Duisburg47048 Duisburg, Germany, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A semiprime segment of a ring R is a pair P2 ⊂ P1 of semiprime ideals of R such that ∩ In ⊆ P2 for all ideals I of R with P2 ⊂ I ⊂ P1. In this paper semiprime segments with P1 a comparizer ideal are classified as either simple, exceptional, or archimedean, extending to several classes of rings a classification known for right chain rings. These three types of semiprime segments are also characterized in terms of the pseudo-radical.

MSC classification

Secondary: 16D70: Structure and classification (except as in ), direct sum decomposition, cancellation 16W99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 80 , Issue 2 , April 2006 , pp. 263 - 272
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Auslander, M., Green, E. L. and Reiten, I., ‘Modules with waists’, Illinois J. Math. 19 (1975), 467–478.CrossRefGoogle Scholar
[2]Badawi, A., Anderson, D. F. and Dobbs, D. E., ‘Pseudo-valuation rings’, in: Commutative ring theory (Fés, 1995), Lecture Notes in Pure and Appl. Math. 185 (Dekker, New York, 1997) pp. 57–67.Google Scholar
[3]Brungs, H. H., Marubayashi, H. and Osmanagic, E., ‘A classification of prime segments in simple artinian rings’, Proc. Amer. Math. Soc. 128 (2000), 3167–3175.CrossRefGoogle Scholar
[4]Brungs, H. H. and Schröder, M., ‘Prime segments of skew fields’, Canad. J. Math. 47 (1995), 1148–1176.CrossRefGoogle Scholar
[5]Brungs, H. H. and Törner, G., ‘Ideal theory of right cones and associated rings’, J. Algebra 210 (1998), 145–164.CrossRefGoogle Scholar
[6]Dubrovin, N. I., ‘Chain domains’, Moscow Univ. Math. Bull. 36 (1980), 56–60.Google Scholar
[7]Dubrovin, N. I., ‘Rational closures of group rings of left-ordered groups’, Mat. Sb. 184 (1993), 3–48 (Russian).Google Scholar
[8]Ferrero, M. and Sant'Ana, A., ‘Rings with comparability’, Canad. Math. Bull. 42 (1999), 174–183.CrossRefGoogle Scholar
[9]Ferrero, M. and Törner, G., ‘On the ideal structure of right distributive rings’, Comm. Algebra 21 (1993), 2697–2713.CrossRefGoogle Scholar
[10]Ferrero, M. and Törner, G., ‘On waists of right distributive rings’, Forum Math. 7 (1995), 419–433.CrossRefGoogle Scholar
[11]Gilmer, R., Multiplicative ideal theory, Pure and Appl. Math. 12 (Marcel Dekker, New York, 1972).Google Scholar
[12]Mathiak, K., ‘Zur Bewertungstheorie nicht kommutativer Körper’, J. Algebra 73 (1981), 586–600.CrossRefGoogle Scholar
[13]Mazurek, R., ‘Distributive rings with Goldie dimension one’, Comm. Algebra 19 (1991), 931–944.CrossRefGoogle Scholar
[14]Mazurek, R., ‘Pseudo-chain rings and pseudo-uniserial modules’, Comm. Algebra 33 (2005), 1519–1527.CrossRefGoogle Scholar
[15]Mazurek, R. and Törner, G., ‘Comparizer ideals of rings’, Comm. Algebra 32 (2004), 4653–4665.CrossRefGoogle Scholar
[16]Stephenson, W., ‘Modules whose lattice of submodules is distributive’, Proc. London Math. Soc. 28 (1974), 291–310.CrossRefGoogle Scholar