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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]
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Abstract

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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

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