. . . . . . . . . "\uCD94\uC0C1\uB300\uC218\uD559\uC5D0\uC11C \uAC12\uB9E4\uAE40\uD658(-\u74B0, \uC601\uC5B4: valuation ring) \uB610\uB294 \uBD80\uCE58\uD658(\u8CE6\u5024\u74B0)\uC740 \uC815\uC218\uC758 \uD658\uC758 \uAD6D\uC18C\uD654 \uC640 \uC720\uC0AC\uD55C \uC131\uC9C8\uC744 \uAC00\uC9C0\uB294 \uC815\uC5ED\uC774\uB2E4."@ko . . . . . . . . . . "In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x\u22121 belongs to D. Given a field F, if D is a subring of F such that either x or x\u22121 belongs toD for every nonzero x in F, then D is said to be a valuation ring for the field F or a place of F. Since F in this case is indeed the field of fractions of D, a valuation ring for a field is a valuation ring. Another way to characterize the valuation rings of a field F is that valuation rings D of F have F as their field of fractions, and their ideals are totally ordered by inclusion; or equivalently their principal ideals are totally ordered by inclusion. In particular, every valuation ring is a local ring."@en . . "In algebra, un anello di valutazione (o dominio di valutazione) \u00E8 un anello commutativo unitario integro A tale che, per ogni x nel suo campo dei quozienti, almeno uno tra e \u00E8 in A; equivalentemente, \u00E8 un anello commutativo integro i cui ideali sono totalmente ordinati. Esempi di anelli di valutazione sono le localizzazioni di e di (dove K \u00E8 un campo) su un loro ideale primo, oppure l'anello degli interi p-adici per un numero primo p, o ancora l'anello delle serie formali su un campo. Una versione \"globale\" degli anelli di valutazione sono i domini di Pr\u00FCfer, che sono quegli anelli in cui, per ogni ideale primo P, la localizzazione AP \u00E8 un anello di valutazione."@it . . "23593"^^ . "966856"^^ . . "\u0423 \u043A\u043E\u043C\u0443\u0442\u0430\u0442\u0438\u0432\u043D\u0456\u0439 \u0430\u043B\u0433\u0435\u0431\u0440\u0456 \u043A\u0456\u043B\u044C\u0446\u0435\u043C \u043D\u043E\u0440\u043C\u0443\u0432\u0430\u043D\u043D\u044F \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u043E\u0431\u043B\u0430\u0441\u0442\u044C \u0446\u0456\u043B\u0456\u0441\u043D\u043E\u0441\u0442\u0456, \u0449\u043E \u0437\u0430\u0434\u043E\u0432\u043E\u043B\u044C\u043D\u044F\u0454 \u0434\u0435\u044F\u043A\u0438\u043C \u0434\u043E\u0434\u0430\u0442\u043A\u043E\u0432\u0438\u043C \u0432\u0438\u043C\u043E\u0433\u0430\u043C. \u041A\u0456\u043B\u044C\u0446\u044F \u043D\u043E\u0440\u043C\u0443\u0432\u0430\u043D\u043D\u044F \u0454 \u043F\u043E\u0432'\u044F\u0437\u0430\u043D\u0438\u043C\u0438 \u0437 \u043F\u043E\u043D\u044F\u0442\u0442\u044F\u043C \u043D\u043E\u0440\u043C\u0443\u0432\u0430\u043D\u043D\u044F \u043D\u0430 \u043F\u043E\u043B\u0456. \u041C\u0430\u044E\u0442\u044C \u0448\u0438\u0440\u043E\u043A\u0435 \u0437\u0430\u0441\u0442\u043E\u0441\u0443\u0432\u0430\u043D\u043D\u044F \u0432 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0457\u0447\u043D\u0456\u0439 \u0442\u0435\u043E\u0440\u0456\u0457 \u0447\u0438\u0441\u0435\u043B \u0456 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0457\u0447\u043D\u0456\u0439 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457."@uk . . . . . . . "Valuation ring"@en . "\uCD94\uC0C1\uB300\uC218\uD559\uC5D0\uC11C \uAC12\uB9E4\uAE40\uD658(-\u74B0, \uC601\uC5B4: valuation ring) \uB610\uB294 \uBD80\uCE58\uD658(\u8CE6\u5024\u74B0)\uC740 \uC815\uC218\uC758 \uD658\uC758 \uAD6D\uC18C\uD654 \uC640 \uC720\uC0AC\uD55C \uC131\uC9C8\uC744 \uAC00\uC9C0\uB294 \uC815\uC5ED\uC774\uB2E4."@ko . "\u041A\u0456\u043B\u044C\u0446\u0435 \u043D\u043E\u0440\u043C\u0443\u0432\u0430\u043D\u043D\u044F"@uk . . . "\u5728\u62BD\u8C61\u4EE3\u6578\u4E2D\uFF0C\u8CE6\u503C\u74B0\u662F\u4E00\u500B\u57DF\u88E1\u7684\u4E00\u985E\u7279\u5225\u5B50\u74B0\uFF0C\u53EF\u7531\u57DF\u4E0A\u7684\u67D0\u500B\u8CE6\u503C\u5B9A\u7FA9\u3002\u96E2\u6563\u8CE6\u503C\u74B0\u662F\u5176\u4E2D\u8F03\u5BB9\u6613\u64CD\u4F5C\u7684\u4E00\u985E\u3002"@zh . "Valuatiering"@nl . . . . . . . "\u0423 \u043A\u043E\u043C\u0443\u0442\u0430\u0442\u0438\u0432\u043D\u0456\u0439 \u0430\u043B\u0433\u0435\u0431\u0440\u0456 \u043A\u0456\u043B\u044C\u0446\u0435\u043C \u043D\u043E\u0440\u043C\u0443\u0432\u0430\u043D\u043D\u044F \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u043E\u0431\u043B\u0430\u0441\u0442\u044C \u0446\u0456\u043B\u0456\u0441\u043D\u043E\u0441\u0442\u0456, \u0449\u043E \u0437\u0430\u0434\u043E\u0432\u043E\u043B\u044C\u043D\u044F\u0454 \u0434\u0435\u044F\u043A\u0438\u043C \u0434\u043E\u0434\u0430\u0442\u043A\u043E\u0432\u0438\u043C \u0432\u0438\u043C\u043E\u0433\u0430\u043C. \u041A\u0456\u043B\u044C\u0446\u044F \u043D\u043E\u0440\u043C\u0443\u0432\u0430\u043D\u043D\u044F \u0454 \u043F\u043E\u0432'\u044F\u0437\u0430\u043D\u0438\u043C\u0438 \u0437 \u043F\u043E\u043D\u044F\u0442\u0442\u044F\u043C \u043D\u043E\u0440\u043C\u0443\u0432\u0430\u043D\u043D\u044F \u043D\u0430 \u043F\u043E\u043B\u0456. \u041C\u0430\u044E\u0442\u044C \u0448\u0438\u0440\u043E\u043A\u0435 \u0437\u0430\u0441\u0442\u043E\u0441\u0443\u0432\u0430\u043D\u043D\u044F \u0432 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0457\u0447\u043D\u0456\u0439 \u0442\u0435\u043E\u0440\u0456\u0457 \u0447\u0438\u0441\u0435\u043B \u0456 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0457\u0447\u043D\u0456\u0439 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457."@uk . . . . "In de commutatieve algebra, een tak van de hogere wiskunde, is een valuatiering een bijzonder soort commutatieve ring met eenheidselement."@nl . . . . . "Anello di valutazione"@it . . . . . . . . . . . "In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x\u22121 belongs to D. Given a field F, if D is a subring of F such that either x or x\u22121 belongs toD for every nonzero x in F, then D is said to be a valuation ring for the field F or a place of F. Since F in this case is indeed the field of fractions of D, a valuation ring for a field is a valuation ring. Another way to characterize the valuation rings of a field F is that valuation rings D of F have F as their field of fractions, and their ideals are totally ordered by inclusion; or equivalently their principal ideals are totally ordered by inclusion. In particular, every valuation ring is a local ring. The valuation rings of a field are the maximal elements of the set of the local subrings in the field partially ordered by dominance or refinement, where dominates if and . Every local ring in a field K is dominated by some valuation ring of K. An integral domain whose localization at any prime ideal is a valuation ring is called a Pr\u00FCfer domain."@en . . "\uAC12\uB9E4\uAE40\uD658"@ko . . . . . . . . . . "Valua\u010Dn\u00ED okruh"@cs . . "V oboru abstraktn\u00ED algebry je valua\u010Dn\u00ED okruh takov\u00FD obor integrity , \u017Ee pro ka\u017Ed\u00FD prvek jeho pod\u00EDlov\u00E9ho t\u011Blesa plat\u00ED bu\u010F nebo"@cs . . . "\u62BD\u8C61\u4EE3\u6570\u5B66\u306B\u304A\u3044\u3066\u3001\u4ED8\u5024\u74B0\uFF08\u3075\u3061\u304B\u3093\u3001\u82F1: valuation ring\uFF09\u3068\u306F\u3001\u6574\u57DF D \u3067\u3042\u3063\u3066\u3001\u305D\u306E\u5206\u6570\u4F53 F \u306E\u3059\u3079\u3066\u306E\u5143 x \u306B\u5BFE\u3057\u3066\u3001x \u304B x \u22121 \u306E\u5C11\u306A\u304F\u3068\u3082\u4E00\u65B9\u304C D \u306B\u5C5E\u3059\u308B\u3088\u3046\u306A\u3082\u306E\u3067\u3042\u308B\u3002 \u4F53 F \u304C\u4E0E\u3048\u3089\u308C\u305F\u3068\u304D\u3001D \u304C F \u306E\u90E8\u5206\u74B0\u3067\u3042\u3063\u3066\u3001F \u306E\u3059\u3079\u3066\u306E 0 \u3067\u306A\u3044\u5143 x \u306B\u5BFE\u3057\u3066 x \u304B x \u22121 \u304C D \u306B\u5C5E\u3057\u3066\u3044\u308B\u3068\u304D\u3001D \u3092 \u4F53 F \u306E\u4ED8\u5024\u74B0\uFF08a valuation ring for the field F\uFF09\u307E\u305F\u306F\u5EA7 (place of F) \u3068\u3044\u3046\u3002\u3053\u306E\u5834\u5408 F \u306F\u78BA\u304B\u306B D \u306E\u5206\u6570\u4F53\u3067\u3042\u308B\u306E\u3067\u3001\u4F53\u306E\u4ED8\u5024\u74B0\u306F\u4ED8\u5024\u74B0\u3067\u3042\u308B\u3002\u4F53 F \u306E\u4ED8\u5024\u74B0\u3092\u7279\u5FB4\u3065\u3051\u308B\u5225\u306E\u65B9\u6CD5\u306F\u3001F \u306E\u4ED8\u5024\u74B0 D \u306F F \u3092\u305D\u306E\u5206\u6570\u4F53\u3068\u3057\u3066\u3082\u3061\u3001\u305D\u306E\u30A4\u30C7\u30A2\u30EB\u306F\u5305\u542B\u95A2\u4FC2\u3067\u5168\u9806\u5E8F\u3065\u3051\u3089\u308C\u3066\u3044\u308B\u3001\u3042\u308B\u3044\u306F\u540C\u3058\u3053\u3068\u3060\u304C\u3001\u305D\u306E\u5358\u9805\u30A4\u30C7\u30A2\u30EB\u304C\u5305\u542B\u95A2\u4FC2\u3067\u5168\u9806\u5E8F\u4ED8\u3051\u3089\u308C\u3066\u3044\u308B\u3053\u3068\u3067\u3042\u308B\u3002\u3068\u304F\u306B\u3001\u3059\u3079\u3066\u306E\u4ED8\u5024\u74B0\u306F\u5C40\u6240\u74B0\u3067\u3042\u308B\u3002 \u4F53\u306E\u4ED8\u5024\u74B0\u306F\u652F\u914D\uFF08dominance\uFF09\u3042\u308B\u3044\u306F\u7D30\u5206\uFF08refinement\uFF09\u306B\u3088\u3063\u3066\u9806\u5E8F\u3092\u5165\u308C\u305F\u4F53\u306E\u5C40\u6240\u90E8\u5206\u74B0\u306E\u96C6\u5408\u306E\u6975\u5927\u5143\u3067\u3042\u308B\u3001\u305F\u3060\u3057 \u304B\u3064 \u306A\u3089\u3070\u3001 \u306F \u3092\u652F\u914D\u3059\u308B\u3002 \u4F53 K \u306E\u3059\u3079\u3066\u306E\u5C40\u6240\u74B0\u306F K \u306E\u3042\u308B\u4ED8\u5024\u74B0\u306B\u3088\u3063\u3066\u652F\u914D\u3055\u308C\u308B\u3002 \u4EFB\u610F\u306E\u7D20\u30A4\u30C7\u30A2\u30EB\u306B\u304A\u3051\u308B\u5C40\u6240\u5316\u304C\u4ED8\u5024\u74B0\u3067\u3042\u308B\u3088\u3046\u306A\u6574\u57DF\u306F\u30D7\u30EA\u30E5\u30FC\u30D5\u30A1\u30FC\u6574\u57DF\u3068\u547C\u3070\u308C\u308B\u3002"@ja . "\u8CE6\u503C\u74B0"@zh . . . "In algebra, un anello di valutazione (o dominio di valutazione) \u00E8 un anello commutativo unitario integro A tale che, per ogni x nel suo campo dei quozienti, almeno uno tra e \u00E8 in A; equivalentemente, \u00E8 un anello commutativo integro i cui ideali sono totalmente ordinati. Esempi di anelli di valutazione sono le localizzazioni di e di (dove K \u00E8 un campo) su un loro ideale primo, oppure l'anello degli interi p-adici per un numero primo p, o ancora l'anello delle serie formali su un campo."@it . . "V oboru abstraktn\u00ED algebry je valua\u010Dn\u00ED okruh takov\u00FD obor integrity , \u017Ee pro ka\u017Ed\u00FD prvek jeho pod\u00EDlov\u00E9ho t\u011Blesa plat\u00ED bu\u010F nebo"@cs . . . . . . . . . . . . . . . . . . . "\u4ED8\u5024\u74B0"@ja . . . "In de commutatieve algebra, een tak van de hogere wiskunde, is een valuatiering een bijzonder soort commutatieve ring met eenheidselement."@nl . . . . . . . . . . . . . . . . . . . . . . . . "1087452635"^^ . . . "\u5728\u62BD\u8C61\u4EE3\u6578\u4E2D\uFF0C\u8CE6\u503C\u74B0\u662F\u4E00\u500B\u57DF\u88E1\u7684\u4E00\u985E\u7279\u5225\u5B50\u74B0\uFF0C\u53EF\u7531\u57DF\u4E0A\u7684\u67D0\u500B\u8CE6\u503C\u5B9A\u7FA9\u3002\u96E2\u6563\u8CE6\u503C\u74B0\u662F\u5176\u4E2D\u8F03\u5BB9\u6613\u64CD\u4F5C\u7684\u4E00\u985E\u3002"@zh . "\u62BD\u8C61\u4EE3\u6570\u5B66\u306B\u304A\u3044\u3066\u3001\u4ED8\u5024\u74B0\uFF08\u3075\u3061\u304B\u3093\u3001\u82F1: valuation ring\uFF09\u3068\u306F\u3001\u6574\u57DF D \u3067\u3042\u3063\u3066\u3001\u305D\u306E\u5206\u6570\u4F53 F \u306E\u3059\u3079\u3066\u306E\u5143 x \u306B\u5BFE\u3057\u3066\u3001x \u304B x \u22121 \u306E\u5C11\u306A\u304F\u3068\u3082\u4E00\u65B9\u304C D \u306B\u5C5E\u3059\u308B\u3088\u3046\u306A\u3082\u306E\u3067\u3042\u308B\u3002 \u4F53 F \u304C\u4E0E\u3048\u3089\u308C\u305F\u3068\u304D\u3001D \u304C F \u306E\u90E8\u5206\u74B0\u3067\u3042\u3063\u3066\u3001F \u306E\u3059\u3079\u3066\u306E 0 \u3067\u306A\u3044\u5143 x \u306B\u5BFE\u3057\u3066 x \u304B x \u22121 \u304C D \u306B\u5C5E\u3057\u3066\u3044\u308B\u3068\u304D\u3001D \u3092 \u4F53 F \u306E\u4ED8\u5024\u74B0\uFF08a valuation ring for the field F\uFF09\u307E\u305F\u306F\u5EA7 (place of F) \u3068\u3044\u3046\u3002\u3053\u306E\u5834\u5408 F \u306F\u78BA\u304B\u306B D \u306E\u5206\u6570\u4F53\u3067\u3042\u308B\u306E\u3067\u3001\u4F53\u306E\u4ED8\u5024\u74B0\u306F\u4ED8\u5024\u74B0\u3067\u3042\u308B\u3002\u4F53 F \u306E\u4ED8\u5024\u74B0\u3092\u7279\u5FB4\u3065\u3051\u308B\u5225\u306E\u65B9\u6CD5\u306F\u3001F \u306E\u4ED8\u5024\u74B0 D \u306F F \u3092\u305D\u306E\u5206\u6570\u4F53\u3068\u3057\u3066\u3082\u3061\u3001\u305D\u306E\u30A4\u30C7\u30A2\u30EB\u306F\u5305\u542B\u95A2\u4FC2\u3067\u5168\u9806\u5E8F\u3065\u3051\u3089\u308C\u3066\u3044\u308B\u3001\u3042\u308B\u3044\u306F\u540C\u3058\u3053\u3068\u3060\u304C\u3001\u305D\u306E\u5358\u9805\u30A4\u30C7\u30A2\u30EB\u304C\u5305\u542B\u95A2\u4FC2\u3067\u5168\u9806\u5E8F\u4ED8\u3051\u3089\u308C\u3066\u3044\u308B\u3053\u3068\u3067\u3042\u308B\u3002\u3068\u304F\u306B\u3001\u3059\u3079\u3066\u306E\u4ED8\u5024\u74B0\u306F\u5C40\u6240\u74B0\u3067\u3042\u308B\u3002 \u4F53\u306E\u4ED8\u5024\u74B0\u306F\u652F\u914D\uFF08dominance\uFF09\u3042\u308B\u3044\u306F\u7D30\u5206\uFF08refinement\uFF09\u306B\u3088\u3063\u3066\u9806\u5E8F\u3092\u5165\u308C\u305F\u4F53\u306E\u5C40\u6240\u90E8\u5206\u74B0\u306E\u96C6\u5408\u306E\u6975\u5927\u5143\u3067\u3042\u308B\u3001\u305F\u3060\u3057 \u304B\u3064 \u306A\u3089\u3070\u3001 \u306F \u3092\u652F\u914D\u3059\u308B\u3002 \u4F53 K \u306E\u3059\u3079\u3066\u306E\u5C40\u6240\u74B0\u306F K \u306E\u3042\u308B\u4ED8\u5024\u74B0\u306B\u3088\u3063\u3066\u652F\u914D\u3055\u308C\u308B\u3002 \u4EFB\u610F\u306E\u7D20\u30A4\u30C7\u30A2\u30EB\u306B\u304A\u3051\u308B\u5C40\u6240\u5316\u304C\u4ED8\u5024\u74B0\u3067\u3042\u308B\u3088\u3046\u306A\u6574\u57DF\u306F\u30D7\u30EA\u30E5\u30FC\u30D5\u30A1\u30FC\u6574\u57DF\u3068\u547C\u3070\u308C\u308B\u3002"@ja . . .