A Seminar on Multiplication Modules By Turki Al-Suriheed.

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Presentation transcript:

A Seminar on Multiplication Modules By Turki Al-Suriheed

Key Definitions (1) Modules

( 2 ) Submodules

Lemma

( 3 ) The ideal [ N:M ]

In the following all rings are commutative with identity and all modules are unital modules.

The Abstract

The Idea of Multiplication Modules

So, the concept of multiplication rings was given to generalize the concept of Dedekinds domain.

Is any R-module multiplication module?

Examples ## Any cyclic module is multiplication. ## Any invertible ideal is multiplication ideal. ## Any ideal of Von Neumann Regular ring is multiplication ideal.

Theorem

The Aim If M is a faithful multiplicatin R-module and N,K are submodules of M then NK is Possible.

Let M be a faithful multiplication module

Let S={ A : A an ideal of R such that M=AM } Define

Theorem If M is a faithful multiplication R-module then, ( 1 ) M=T(M)M. ( 2 ) T(M)=

Question ? Let M be an R-module and A, B ideals of R

Cancellation Law

Corollary

The Main Result

THE END