1. A Seminar on Multiplication Modules By Turki Al-Suriheed

2. Key Definitions (1) Modules

4. ( 2 ) Submodules

5. Lemma

6. ( 3 ) The ideal [ N:M ]

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

9. The Abstract

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?

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.

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

27. Cancellation Law

28. Corollary

29. The Main Result

31. THE END