IJPAM: Volume 20, No. 1 (2005)

HILBERT MODULES

M. Maani Shirazi$^1$, H. Sharif$^2$
$^{1,2}$Department of Mathematics
Shiraz University
Shiraz, 71454, IRAN
$^1$email: liliumsor7@yahoo.com
$^2$email: sharif@math.susc.ac.ir


Abstract.We shall extend the notion of Hilbert rings to Hilbert modules. We call an $R$-module $M$ a Hilbert module if every prime submodule of $M$ is the intersection of all the maximal submodules containing it. It will be shown that every finitely generated module over a ring $R$ is a Hilbert module if and only if $R$ is a Hilbert ring. Moreover, we prove that a finitely generated $R$-module $M$ is a Hilbert module if and only if $R/\text{Ann}\,_{R}(M)$ is a Hilbert ring.

Received: October 12, 2004

AMS Subject Classification: 13C05, 16D80

Key Words and Phrases: Hilbert modules, Hilbert rings

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 20
Issue: 1