Skip to main content
MathOverflow

Return to Answer

added 16 characters in body
Source Link
R.P.
R.P.
  • 4.7k
  • 19
  • 43
  • 67

Only in principal ideal domains (PIDs). If by number ring you mean Dedekind domain, then all its localizations at prime ideals are discrete valuation rings (except the one at 0 which is a field), which are principal ideal domains. So then every ideal in a Dedekind Domain is locally principal. But of course there are Dedekind domains that are not PIDs!

Only in principal ideal domains (PIDs). If by number ring you mean Dedekind domain, then all its localizations at prime ideals are discrete valuation rings (except the one at 0 which is a field), which are principal ideal domains. So then every ideal is locally principal. But of course there are Dedekind domains that are not PIDs!

Only in principal ideal domains (PIDs). If by number ring you mean Dedekind domain, then all its localizations at prime ideals are discrete valuation rings (except the one at 0 which is a field), which are principal ideal domains. So every ideal in a Dedekind Domain is locally principal. But of course there are Dedekind domains that are not PIDs!

Source Link
R.P.
R.P.
  • 4.7k
  • 19
  • 43
  • 67

Only in principal ideal domains (PIDs). If by number ring you mean Dedekind domain, then all its localizations at prime ideals are discrete valuation rings (except the one at 0 which is a field), which are principal ideal domains. So then every ideal is locally principal. But of course there are Dedekind domains that are not PIDs!