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A Seminar on Multiplication Modules By Turki Al-Suriheed.

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Presentation on theme: "A Seminar on Multiplication Modules By Turki Al-Suriheed."— Presentation transcript:

1 A Seminar on Multiplication Modules By Turki Al-Suriheed

2 Key Definitions (1) Modules

3

4 ( 2 ) Submodules

5 Lemma

6 ( 3 ) The ideal [ N:M ]

7

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

9 The Abstract

10

11 The Idea of Multiplication Modules

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

13 Is any R-module multiplication module?

14

15

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

17 Theorem

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

19

20

21

22 Let M be a faithful multiplication module

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

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

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

26

27 Cancellation Law

28 Corollary

29 The Main Result

30

31 THE END


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