# The Transform Formula for Submodules of Multiplication Modules

### New Zealand Journal of Mathematics

Vol. 41, (2011), Pages 25-37

Majid M. Ali

Department of Mathematics and Statistics

Sultan Qaboos University

Muscat, Oman

Abstract Let R be an integral domain with quotient field Q(R) and M a faithful multiplication R-module. For a submodule N, let where

$B_{n}=\left\{ x\in Q\left( R\right) :x\left[ N:M\right] ^{n}N\subseteq M\right.$ for some positive integer $\left. n\right\}.$

We study the problem of determining for which multiplication modules over integral domains we have the equality $T\left( \left[ K:M\right] N\right) =T\left( K\right) +T\left( N\right)$ for all submodules, or all finitely generated submodules, or all cyclic submodules K and N of M. \ We show that a faithful multiplication Prufer module over an integral domain satisfies $T\left( \left[ K:M\right] N\right) =T\left( K\right) +T\left( N\right)$ for all finitely generated submodules K and N of M.

Keywords Transform formula, Multiplication module, Prufer module.

Classification (MSC2000) 13C13, 13A15.