Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-29T08:27:21.300Z Has data issue: false hasContentIssue false
  • English
  • Français

BOUNDED AND FULLY BOUNDED MODULES

Part of: Chain conditions, growth conditions, and other forms of finiteness

Published online by Cambridge University Press:  04 October 2011

A. HAGHANY ,
M. MAZROOEI  and
M. R. VEDADI
A. HAGHANY
Affiliation:
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran (email: [email protected])
M. MAZROOEI
Affiliation:
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran (email: [email protected])
M. R. VEDADI*
Affiliation:
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran (email: [email protected])
*
✉For correspondence; 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.

Generalizing the concept of right bounded rings, a module MR is called bounded if annR(M/N)≤eRR for all N≤eMR. The module MR is called fully bounded if (M/P) is bounded as a module over R/annR(M/P) for any ℒ2-prime submodule P◃MR. Boundedness and right boundedness are Morita invariant properties. Rings with all modules (fully) bounded are characterized, and it is proved that a ring R is right Artinian if and only if RR has Krull dimension, all R-modules are fully bounded and ideals of R are finitely generated as right ideals. For certain fully bounded ℒ2-Noetherian modules MR, it is shown that the Krull dimension of MR is at most equal to the classical Krull dimension of R when both dimensions exist.

Keywords

bounded moduleFBN ringKrull dimensionℒ2-Noetherian moduleℒ2-prime module

MSC classification

Secondary: 16P60: Chain conditions on annihilators and summands 16P70: Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 3 , December 2011 , pp. 433 - 440
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Anderson, F. W. and Fuller, K. R., Rings and Categories of Modules, Granduate Texts in Mathematics 13 (Springer, Berlin, 1973).Google Scholar
[2]Bican, L., Jambor, P., Kepka, T. and Nemec, P., ‘Prime and coprime modules’, Fund. Math. 57 (1980), 33–45.CrossRefGoogle Scholar
[3]Goodearl, K. R. and Warfield, R. B. Jr, An Introduction to Noncommutative Noetherian Rings (Cambridge University Press, New York, 2004).CrossRefGoogle Scholar
[4]Haghany, A. and Vedadi, M. R., ‘Endoprime Modules’, Acta Math. Hungar. 106(1–2) (2005), 89–99.CrossRefGoogle Scholar
[5]Heakyung, L., ‘Strongly right FBN rings’, Bull. Aust. Math. Soc. 38(3) (1988), 457–464.Google Scholar
[6]Heakyung, L., ‘Right FBN rings and bounded modules’, Comm. Algebra 16(5) (1988), 977–987.Google Scholar
[7]Heakyung, L., ‘On relatively FBN rings’, Comm. Algebra 23(8) (1995), 2991–3001.Google Scholar
[8]Kok-Ming, T., ‘Homological properties of fully bounded Noetherian rings’, J. Lond. Math. Soc. 2(55)(1) (1997), 37–54.Google Scholar
[9]McConnell, J. C. and Robson, J. C., Noncommutative Noetherian Rings (Wiley-Interscience, New York, 1987).Google Scholar
[10]Smith, P. F., ‘Modules with many homomorphisms’, J. Pure Appl. Algebra 197(1–3) (2005), 305–321.CrossRefGoogle Scholar
[11]Vedadi, M. R., ‘ℒ2-prime and dimensional modules’, Int. Electron. J. Algebra 7 (2010), 47–58.Google Scholar
[12]Wijayanti, I. E. and Wisbauer, R., ‘On coprime modules and comodules’, Comm. Algebra 37(4) (2009), 1308–1333.CrossRefGoogle Scholar