More Results on Almost Noetherian Domains

Adil   Jabbar

doi:10.24996/ijs.2009.50.Appendix.%g

More Results on Almost Noetherian Domains

Authors

Adil Jabbar Department of Mathematics, College of Science, University of Sulaimani, Sulaimani-Iraq.

DOI:

https://doi.org/10.24996/ijs.2009.50.Appendix.%25g

Keywords:

Almost, Domains

Abstract

In this paper we prove some theorems, the first states: If R is an almost
Noetherian domain, then the following statements are equivalent: ١- R is an almost
Dedekind domain.
٢- A(B∩C)= AB∩AC, for all ideals A, B and C of R. ٣- (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: ١- R is a valuation
domain. ٢- The nonunits of R form a nonzero principal ideal of R. ٣- R is integrally
closed and has exactly one nonzero proper prime ideal. In addition to the above
some other results are proved.