. "\uAC00\uD658\uB300\uC218\uD559\uC5D0\uC11C \uACE0\uB7F0\uC2A4\uD2F4 \uD658(Gorenstein\u74B0, \uC601\uC5B4: Gorenstein ring)\uC740 \uAD6D\uC18C\uC801\uC73C\uB85C \uD45C\uC900 \uC120\uB2E4\uBC1C\uC758 \uB2E8\uBA74\uC758 \uAC00\uAD70\uCE35\uC774 \uC790\uC720 \uAC00\uAD70\uCE35\uC778 \uAC00\uD658\uD658\uC774\uB2E4.:519 \uC989, \uD2B9\uC774\uC810\uC744 \uAC00\uC9C8 \uC218 \uC788\uC9C0\uB9CC, \uD2B9\uC774\uC810\uC774 \uBE44\uAD50\uC801\uC73C\uB85C \"\uC815\uCE59\uC801\uC778\" \uC544\uD540 \uC2A4\uD0B4\uC5D0 \uB300\uC751\uD558\uB294 \uAC00\uD658\uD658\uC774\uB2E4."@ko . . . . . "\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 \u0413\u043E\u0440\u0435\u043D\u0448\u0442\u0435\u0439\u043D\u0430 \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u043A\u043E\u043C\u0443\u0442\u0430\u0442\u0438\u0432\u043D\u0435 \u043D\u0435\u0442\u0435\u0440\u043E\u0432\u0435 \u043B\u043E\u043A\u0430\u043B\u044C\u043D\u0435 \u043A\u0456\u043B\u044C\u0446\u0435, \u0449\u043E \u043C\u0430\u0454 \u0441\u043A\u0456\u043D\u0447\u0435\u043D\u043D\u0443 \u0456\u043D'\u0454\u043A\u0442\u0438\u0432\u043D\u0443 \u0440\u043E\u0437\u043C\u0456\u0440\u043D\u0456\u0441\u0442\u044C. \u0411\u0456\u043B\u044C\u0448 \u0437\u0430\u0433\u0430\u043B\u044C\u043D\u043E \u043D\u0435\u0442\u0435\u0440\u043E\u0432\u0435 \u043A\u0456\u043B\u044C\u0446\u0435 \u0430\u0431\u043E \u0441\u0445\u0435\u043C\u0430 \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u043A\u0456\u043B\u044C\u0446\u0435\u043C (\u0441\u0445\u0435\u043C\u043E\u044E) \u0413\u043E\u0440\u0435\u043D\u0448\u0442\u0435\u0439\u043D\u0430, \u044F\u043A\u0449\u043E \u0432\u0441\u0456 \u043B\u043E\u043A\u0430\u043B\u0456\u0437\u0430\u0446\u0456\u0457 \u0446\u044C\u043E\u0433\u043E \u043A\u0456\u043B\u044C\u0446\u044F \u0437\u0430 \u043F\u0440\u043E\u0441\u0442\u0438\u043C\u0438 \u0456\u0434\u0435\u0430\u043B\u0430\u043C\u0438 (\u0432\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u043D\u043E \u0432\u0441\u0456 \u043B\u043E\u043A\u0430\u043B\u044C\u043D\u0456 \u043A\u0456\u043B\u044C\u0446\u044F \u0441\u0445\u0435\u043C\u0438) \u0454 \u043B\u043E\u043A\u0430\u043B\u044C\u043D\u0438\u043C\u0438 \u043A\u0456\u043B\u044C\u0446\u044F\u043C\u0438 \u0413\u043E\u0440\u0435\u043D\u0448\u0442\u0435\u0439\u043D\u0430."@uk . . "Ein Gorensteinring ist ein Ring, der in der kommutativen Algebra, einem Teilgebiet der Mathematik, untersucht wird. Ein Gorensteinring ist ein Cohen-Macaulay-Ring mit bestimmten zus\u00E4tzlichen Eigenschaften. Eine Gorensteinsingularit\u00E4t ist eine Singularit\u00E4t, deren lokaler Ring ein Gorensteinring ist. Benannt wurden die Ringe nach Daniel Gorenstein, obwohl dieser immer behauptete, nicht einmal die Definition zu verstehen. Dieser Artikel besch\u00E4ftigt sich mit kommutativer Algebra. Insbesondere sind alle betrachteten Ringe kommutativ und haben ein Einselement. Ringhomomorphismen bilden Einselemente auf Einselemente ab. F\u00FCr weitere Details siehe Kommutative Algebra."@de . "\uACE0\uB7F0\uC2A4\uD2F4 \uD658"@ko . . . . . . "1119312286"^^ . "\u845B\u4F96\u65AF\u5766\u74B0"@zh . . . "\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 \u0413\u043E\u0440\u0435\u043D\u0448\u0442\u0435\u0439\u043D\u0430 \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u043A\u043E\u043C\u0443\u0442\u0430\u0442\u0438\u0432\u043D\u0435 \u043D\u0435\u0442\u0435\u0440\u043E\u0432\u0435 \u043B\u043E\u043A\u0430\u043B\u044C\u043D\u0435 \u043A\u0456\u043B\u044C\u0446\u0435, \u0449\u043E \u043C\u0430\u0454 \u0441\u043A\u0456\u043D\u0447\u0435\u043D\u043D\u0443 \u0456\u043D'\u0454\u043A\u0442\u0438\u0432\u043D\u0443 \u0440\u043E\u0437\u043C\u0456\u0440\u043D\u0456\u0441\u0442\u044C. \u0411\u0456\u043B\u044C\u0448 \u0437\u0430\u0433\u0430\u043B\u044C\u043D\u043E \u043D\u0435\u0442\u0435\u0440\u043E\u0432\u0435 \u043A\u0456\u043B\u044C\u0446\u0435 \u0430\u0431\u043E \u0441\u0445\u0435\u043C\u0430 \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u043A\u0456\u043B\u044C\u0446\u0435\u043C (\u0441\u0445\u0435\u043C\u043E\u044E) \u0413\u043E\u0440\u0435\u043D\u0448\u0442\u0435\u0439\u043D\u0430, \u044F\u043A\u0449\u043E \u0432\u0441\u0456 \u043B\u043E\u043A\u0430\u043B\u0456\u0437\u0430\u0446\u0456\u0457 \u0446\u044C\u043E\u0433\u043E \u043A\u0456\u043B\u044C\u0446\u044F \u0437\u0430 \u043F\u0440\u043E\u0441\u0442\u0438\u043C\u0438 \u0456\u0434\u0435\u0430\u043B\u0430\u043C\u0438 (\u0432\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u043D\u043E \u0432\u0441\u0456 \u043B\u043E\u043A\u0430\u043B\u044C\u043D\u0456 \u043A\u0456\u043B\u044C\u0446\u044F \u0441\u0445\u0435\u043C\u0438) \u0454 \u043B\u043E\u043A\u0430\u043B\u044C\u043D\u0438\u043C\u0438 \u043A\u0456\u043B\u044C\u0446\u044F\u043C\u0438 \u0413\u043E\u0440\u0435\u043D\u0448\u0442\u0435\u0439\u043D\u0430."@uk . . "In matematica, in particolare in algebra commutativa, un anello di Gorenstein \u00E8 un anello commutativo tale che la localizzazione in ogni ideale primo \u00E8 un anello di Gorenstein locale. Un anello di Gorenstein locale \u00E8 un anello locale, commutativo, noetheriano R tale che la sua dimensione iniettiva come R-modulo \u00E8 finita. Il concetto di anello di Gorenstein \u00E8 un caso particolare del pi\u00F9 generale concetto di anello di Cohen-Macaulay. Gli analoghi non commutativi degli anelli di Gorenstein di dimensione 0 sono detti ."@it . . . . . . . . . . "In matematica, in particolare in algebra commutativa, un anello di Gorenstein \u00E8 un anello commutativo tale che la localizzazione in ogni ideale primo \u00E8 un anello di Gorenstein locale. Un anello di Gorenstein locale \u00E8 un anello locale, commutativo, noetheriano R tale che la sua dimensione iniettiva come R-modulo \u00E8 finita. Il concetto di anello di Gorenstein \u00E8 un caso particolare del pi\u00F9 generale concetto di anello di Cohen-Macaulay. Gli analoghi non commutativi degli anelli di Gorenstein di dimensione 0 sono detti ."@it . . "Gorenstein ring"@en . . "Gorensteinring"@de . . . . . . . . . "In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring R with finite injective dimension as an R-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring is self-dual in some sense. Frobenius rings are noncommutative analogs of zero-dimensional Gorenstein rings. Gorenstein schemes are the geometric version of Gorenstein rings. For Noetherian local rings, there is the following chain of inclusions. Universally catenary rings \u2283 Cohen\u2013Macaulay rings \u2283 \u2283 complete intersection rings \u2283 regular local rings"@en . . . . . . . "11562"^^ . . "\u53EF\u63DB\u74B0\u8AD6\u306B\u304A\u3044\u3066\u3001Gorenstein \u5C40\u6240\u74B0 (Gorenstein local ring) \u306F\u30CD\u30FC\u30BF\u30FC\u53EF\u63DB\u5C40\u6240\u74B0 R \u3067\u3042\u3063\u3066\u3001R-\u52A0\u7FA4\u3068\u3057\u3066\u6709\u9650\u306E\u79FB\u5165\u6B21\u5143\u3092\u3082\u3064\u3082\u306E\u3067\u3042\u308B\u3002\u540C\u5024\u306A\u6761\u4EF6\u304C\u305F\u304F\u3055\u3093\u3042\u308A\u3001\u305D\u306E\u3046\u3061\u306E\u3044\u304F\u3064\u304B\u306F\u4EE5\u4E0B\u306B\u30EA\u30B9\u30C8\u3055\u308C\u308B\u304C\u3001\u591A\u304F\u306F\u3042\u308B\u7A2E\u306E\u53CC\u5BFE\u306E\u6761\u4EF6\u3092\u6271\u3046\u3002 Gorenstein \u74B0\u306F Grothendieck \u306B\u3088\u3063\u3066\u5C0E\u5165\u3055\u308C\u3001\u5F7C\u304C\u540D\u524D\u3092\u4ED8\u3051\u305F\u304C\u3001\u305D\u306E\u7406\u7531\u306F \u306B\u3088\u3063\u3066\u7814\u7A76\u3055\u308C\u305F\u7279\u7570\u5E73\u9762\u66F2\u7DDA\u306E\u53CC\u5BFE\u306E\u6027\u8CEA\u3068\u306E\u95A2\u4FC2\u3067\u3042\u308B\uFF08Gorenstein \u306F Gorenstein \u74B0\u306E\u5B9A\u7FA9\u3092\u7406\u89E3\u3057\u3066\u3044\u306A\u3044\u3068\u4E3B\u5F35\u3059\u308B\u3053\u3068\u3092\u597D\u3093\u3060\uFF09\u30020\u6B21\u5143\u306E\u30B1\u30FC\u30B9\u306F \u306B\u3088\u3063\u3066\u7814\u7A76\u3055\u308C\u3066\u3044\u305F\u3002 \u3068 \u306F Gorenstein \u74B0\u306E\u6982\u5FF5\u3092\u516C\u8868\u3057\u305F\u3002 0\u6B21\u5143 Gorenstein \u74B0\u306E\u975E\u53EF\u63DB\u74B0\u306B\u304A\u3051\u308B\u985E\u4F3C\u306F\u30D5\u30ED\u30D9\u30CB\u30A6\u30B9\u74B0\u3068\u547C\u3070\u308C\u308B\u3002 \u30CD\u30FC\u30BF\u30FC\u5C40\u6240\u74B0\u306B\u3064\u3044\u3066\u306F\u6B21\u306E\u5305\u542B\u95A2\u4FC2\u304C\u6210\u308A\u7ACB\u3064\u3002 \u5F37\u9396\u72B6\u74B0 \u2283 \u30B3\u30FC\u30A8\u30F3\u30FB\u30DE\u30B3\u30FC\u30EC\u30FC\u74B0 \u2283 \u30B4\u30EC\u30F3\u30B7\u30E5\u30BF\u30A4\u30F3\u74B0 \u2283 \u5B8C\u5168\u4EA4\u53C9\u74B0 \u2283 \u6B63\u5247\u5C40\u6240\u74B0"@ja . . . . . . . . "515613"^^ . . . "Anello di Gorenstein"@it . . . . . . . . . . . . "\u30B4\u30EC\u30F3\u30B7\u30E5\u30BF\u30A4\u30F3\u74B0"@ja . "\u041A\u0456\u043B\u044C\u0446\u0435 \u0413\u043E\u0440\u0435\u043D\u0448\u0442\u0435\u0439\u043D\u0430"@uk . "\u5728\u4EA4\u63DB\u4EE3\u6578\u4E2D\uFF0C\u4E00\u500B\u845B\u4F96\u65AF\u5766\u5C40\u90E8\u74B0\u662F\u4E00\u500B\u5167\u5C04\u7DAD\u5EA6\u6709\u9650\u7684\u4EA4\u63DB\u3001\u5C40\u90E8\u8AFE\u7279\u74B0\u3002\u4E00\u500B\u845B\u4F96\u65AF\u5766\u74B0\uFF08\u82F1\u6587\uFF1AGorenstein ring\uFF09\u662F\u5C0D\u6BCF\u500B\u7D20\u7406\u60F3\u7684\u5C40\u90E8\u5316\u7686\u70BA\u845B\u4F96\u65AF\u5766\u5C40\u90E8\u74B0\u7684\u4EA4\u63DB\u74B0\u3002\u845B\u4F96\u65AF\u5766\u74B0\u662F\u79D1\u6069-\u9EA5\u8003\u5229\u74B0\u7684\u7279\u4F8B\uFF0C\u5B83\u8207\u51DD\u805A\u5C0D\u5076\u6027\u5B9A\u7406\uFF08\u585E\u723E\u5C0D\u5076\u6027\u5B9A\u7406\u7684\u63A8\u5EE3\uFF09\u6709\u5BC6\u5207\u95DC\u4FC2\u3002 \u845B\u4F96\u65AF\u5766\u74B0\u4EE5\u6578\u5B78\u5BB6\u547D\u540D\u3002"@zh . . . . "Ein Gorensteinring ist ein Ring, der in der kommutativen Algebra, einem Teilgebiet der Mathematik, untersucht wird. Ein Gorensteinring ist ein Cohen-Macaulay-Ring mit bestimmten zus\u00E4tzlichen Eigenschaften. Eine Gorensteinsingularit\u00E4t ist eine Singularit\u00E4t, deren lokaler Ring ein Gorensteinring ist. Benannt wurden die Ringe nach Daniel Gorenstein, obwohl dieser immer behauptete, nicht einmal die Definition zu verstehen."@de . "\uAC00\uD658\uB300\uC218\uD559\uC5D0\uC11C \uACE0\uB7F0\uC2A4\uD2F4 \uD658(Gorenstein\u74B0, \uC601\uC5B4: Gorenstein ring)\uC740 \uAD6D\uC18C\uC801\uC73C\uB85C \uD45C\uC900 \uC120\uB2E4\uBC1C\uC758 \uB2E8\uBA74\uC758 \uAC00\uAD70\uCE35\uC774 \uC790\uC720 \uAC00\uAD70\uCE35\uC778 \uAC00\uD658\uD658\uC774\uB2E4.:519 \uC989, \uD2B9\uC774\uC810\uC744 \uAC00\uC9C8 \uC218 \uC788\uC9C0\uB9CC, \uD2B9\uC774\uC810\uC774 \uBE44\uAD50\uC801\uC73C\uB85C \"\uC815\uCE59\uC801\uC778\" \uC544\uD540 \uC2A4\uD0B4\uC5D0 \uB300\uC751\uD558\uB294 \uAC00\uD658\uD658\uC774\uB2E4."@ko . . . . "\u5728\u4EA4\u63DB\u4EE3\u6578\u4E2D\uFF0C\u4E00\u500B\u845B\u4F96\u65AF\u5766\u5C40\u90E8\u74B0\u662F\u4E00\u500B\u5167\u5C04\u7DAD\u5EA6\u6709\u9650\u7684\u4EA4\u63DB\u3001\u5C40\u90E8\u8AFE\u7279\u74B0\u3002\u4E00\u500B\u845B\u4F96\u65AF\u5766\u74B0\uFF08\u82F1\u6587\uFF1AGorenstein ring\uFF09\u662F\u5C0D\u6BCF\u500B\u7D20\u7406\u60F3\u7684\u5C40\u90E8\u5316\u7686\u70BA\u845B\u4F96\u65AF\u5766\u5C40\u90E8\u74B0\u7684\u4EA4\u63DB\u74B0\u3002\u845B\u4F96\u65AF\u5766\u74B0\u662F\u79D1\u6069-\u9EA5\u8003\u5229\u74B0\u7684\u7279\u4F8B\uFF0C\u5B83\u8207\u51DD\u805A\u5C0D\u5076\u6027\u5B9A\u7406\uFF08\u585E\u723E\u5C0D\u5076\u6027\u5B9A\u7406\u7684\u63A8\u5EE3\uFF09\u6709\u5BC6\u5207\u95DC\u4FC2\u3002 \u845B\u4F96\u65AF\u5766\u74B0\u4EE5\u6578\u5B78\u5BB6\u547D\u540D\u3002"@zh . "\u53EF\u63DB\u74B0\u8AD6\u306B\u304A\u3044\u3066\u3001Gorenstein \u5C40\u6240\u74B0 (Gorenstein local ring) \u306F\u30CD\u30FC\u30BF\u30FC\u53EF\u63DB\u5C40\u6240\u74B0 R \u3067\u3042\u3063\u3066\u3001R-\u52A0\u7FA4\u3068\u3057\u3066\u6709\u9650\u306E\u79FB\u5165\u6B21\u5143\u3092\u3082\u3064\u3082\u306E\u3067\u3042\u308B\u3002\u540C\u5024\u306A\u6761\u4EF6\u304C\u305F\u304F\u3055\u3093\u3042\u308A\u3001\u305D\u306E\u3046\u3061\u306E\u3044\u304F\u3064\u304B\u306F\u4EE5\u4E0B\u306B\u30EA\u30B9\u30C8\u3055\u308C\u308B\u304C\u3001\u591A\u304F\u306F\u3042\u308B\u7A2E\u306E\u53CC\u5BFE\u306E\u6761\u4EF6\u3092\u6271\u3046\u3002 Gorenstein \u74B0\u306F Grothendieck \u306B\u3088\u3063\u3066\u5C0E\u5165\u3055\u308C\u3001\u5F7C\u304C\u540D\u524D\u3092\u4ED8\u3051\u305F\u304C\u3001\u305D\u306E\u7406\u7531\u306F \u306B\u3088\u3063\u3066\u7814\u7A76\u3055\u308C\u305F\u7279\u7570\u5E73\u9762\u66F2\u7DDA\u306E\u53CC\u5BFE\u306E\u6027\u8CEA\u3068\u306E\u95A2\u4FC2\u3067\u3042\u308B\uFF08Gorenstein \u306F Gorenstein \u74B0\u306E\u5B9A\u7FA9\u3092\u7406\u89E3\u3057\u3066\u3044\u306A\u3044\u3068\u4E3B\u5F35\u3059\u308B\u3053\u3068\u3092\u597D\u3093\u3060\uFF09\u30020\u6B21\u5143\u306E\u30B1\u30FC\u30B9\u306F \u306B\u3088\u3063\u3066\u7814\u7A76\u3055\u308C\u3066\u3044\u305F\u3002 \u3068 \u306F Gorenstein \u74B0\u306E\u6982\u5FF5\u3092\u516C\u8868\u3057\u305F\u3002 0\u6B21\u5143 Gorenstein \u74B0\u306E\u975E\u53EF\u63DB\u74B0\u306B\u304A\u3051\u308B\u985E\u4F3C\u306F\u30D5\u30ED\u30D9\u30CB\u30A6\u30B9\u74B0\u3068\u547C\u3070\u308C\u308B\u3002 \u30CD\u30FC\u30BF\u30FC\u5C40\u6240\u74B0\u306B\u3064\u3044\u3066\u306F\u6B21\u306E\u5305\u542B\u95A2\u4FC2\u304C\u6210\u308A\u7ACB\u3064\u3002 \u5F37\u9396\u72B6\u74B0 \u2283 \u30B3\u30FC\u30A8\u30F3\u30FB\u30DE\u30B3\u30FC\u30EC\u30FC\u74B0 \u2283 \u30B4\u30EC\u30F3\u30B7\u30E5\u30BF\u30A4\u30F3\u74B0 \u2283 \u5B8C\u5168\u4EA4\u53C9\u74B0 \u2283 \u6B63\u5247\u5C40\u6240\u74B0"@ja . . . . . "In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring R with finite injective dimension as an R-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring is self-dual in some sense. Gorenstein rings were introduced by Grothendieck in his 1961 seminar (published in). The name comes from a duality property of singular plane curves studied by Gorenstein (who was fond of claiming that he did not understand the definition of a Gorenstein ring). The zero-dimensional case had been studied by . and publicized the concept of Gorenstein rings. Frobenius rings are noncommutative analogs of zero-dimensional Gorenstein rings. Gorenstein schemes are the geometric version of Gorenstein rings. For Noetherian local rings, there is the following chain of inclusions. Universally catenary rings \u2283 Cohen\u2013Macaulay rings \u2283 \u2283 complete intersection rings \u2283 regular local rings"@en . . . .