Abstract :
In this paper we prove some theorems, the first states: If R is an almost Noetherian domain, then the following statements are equivalent: 1- R is an almost Dedekind domain. 2- A(B∩C)= AB∩AC, for all ideals A, B and C of R. 3- (A+B)(A∩B)= AB, for all ideals A and B of R and the second states: If R is an almost Noetherian domain which is not a field, then the following statements are equivalent: 1- R is a valuation domain. 2- The nonunits of R form a nonzero principal ideal of R. 3- R is integrally closed and has exactly one nonzero proper prime ideal. In addition to the above some other results are proved.
