you can see this link

http://mate.dm.uba.ar/~aldoc9/Publicaciones/Notas/dedekind.pdf

Thanks Claudia, this paper helps me a lot, the proof is on its last theorem.

 Dear Claudia Pérez Ruisánchez,

May I have some English version, please.

Regards,

Wiwat Wanicharpichat

Dear Wiwat Wanicharpichat,

I dont have the english version of this paper, but you can also see this link. I hope it will be helpful.

best regards

http://www.math.uiuc.edu/~r-ash/Ant/AntChapter3.pdf

Dear Kelvin, dear Claudia, dear Wiwat,

 In most "classical" textbooks in Algebraic Number Theory such as :

Pierre Samuel, "Théorie Algébrique des Nombres", Hermann, Paris, 1967

D. A. Marcus, "Number Fields", Springer Universitext, 1977

the noetherianity property is included in the axioms defining a Dedekind domain. However, in the not less classical book of Cassels & Fröhlich, "Algebraic Number Fields", Academic Press, 1967, it is shown in chapter 1, p. 6, that the axiom "All fractional ideals of D are invertible" (is this the one you take ?) is equivalent to "D is noetherian, integrally closed and all non-zero prime ideals are maximal", and also to "D is noetherian and, for every non-zero prime ideal P of D, the localization of D at P is a discrete valuation ring" .

Best,  Thong

Dear All:

See the book of N. Jacobson, Basic Algebra-II, Section 10.1. Proposition 10.2, is a direct  proof of the Question Posted.   

Regards,

skn

Dear Thong,

Thank you so much for your kindly comments and advice

Regards,

Wiwat