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!
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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! |
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