The Transform Formula for Submodules of Multiplication Modules

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New Zealand Journal of Mathematics

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


Majid M. Ali

Department of Mathematics and Statistics

Sultan Qaboos University

Muscat, Oman

mailto:mali@squ.edu.om






Abstract Let R be an integral domain with quotient field Q(R) and M a faithful multiplication R-module. For a submodule N, let Image:transform1.png 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.

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