proof of localizations of Dedekind domains are Dedekind
is again a Dedekind domain.
We use the characterization of Dedekind domains as integral domains in which every nonzero ideal is invertible (http://planetmath.org/FractionalIdeal) (see proof that a domain is Dedekind if its ideals are invertible).
Let be a nonzero integral ideal of . Then is a nonzero ideal of the Dedekind domain , so it has an inverse
Here, is a fractional ideal of . Also let be the fractional ideal of generated by ,
show that is invertible, so is a Dedekind domain.
|Title||proof of localizations of Dedekind domains are Dedekind|
|Date of creation||2013-03-22 18:35:16|
|Last modified on||2013-03-22 18:35:16|
|Last modified by||gel (22282)|