. . . . . "\u7D20\u7406\u60F3\u5B9A\u7406\uFF08prime ideal theorem\uFF09\u5373\u4FDD\u8BC1\u5728\u7ED9\u5B9A\u7684\u62BD\u8C61\u4EE3\u6570\u4E2D\u7279\u5B9A\u7C7B\u578B\u4E4B\u5B50\u96C6\u7684\u5B58\u5728\u6027\u4E4B\u6578\u5B78\u5B9A\u7406\u3002\u5E38\u89C1\u7684\u4F8B\u5B50\u5C31\u662F\u5E03\u5C14\u7D20\u7406\u60F3\u5B9A\u7406\uFF08Boolean prime ideal theorem\uFF09\uFF0C\u5B83\u58F0\u79F0\u5728\u5E03\u5C14\u4EE3\u6570\u4E2D\u7684\u7406\u60F3\u53EF\u4EE5\u88AB\u6269\u5C55\u6210\u7D20\u7406\u60F3\u3002\u8FD9\u4E2A\u9648\u8FF0\u5BF9\u4E8E\u5728\u96C6\u5408\u4E0A\u7684\u6EE4\u5B50\u7684\u53D8\u4F53\u53EB\u505A\u53EB\u505A\u3002\u901A\u8FC7\u8003\u8651\u4E0D\u540C\u7684\u5E26\u6709\u9002\u5F53\u7684\u7406\u60F3\u6982\u5FF5\u7684\u6570\u5B66\u7ED3\u6784\u53EF\u83B7\u5F97\u5176\u4ED6\u5B9A\u7406\uFF0C\u4F8B\u5982\u74B0\u548C\uFF08\u73AF\u8BBA\u7684\uFF09\u7D20\u7406\u60F3\uFF0C\u548C\u5206\u914D\u683C\u548C\uFF08\u5E8F\u7406\u8BBA\u7684\uFF09\u7684\u6781\u5927\u7406\u60F3\u3002\u672C\u6587\u5173\u6CE8\u5E8F\u7406\u8BBA\u7684\u7D20\u7406\u60F3\u5B9A\u7406\u3002 \u5C3D\u7BA1\u5404\u79CD\u7D20\u7406\u60F3\u5B9A\u7406\u53EF\u80FD\u770B\u8D77\u6765\u7B80\u5355\u4E14\u76F4\u89C9\uFF0C\u5B83\u4EEC\u4E00\u822C\u4E0D\u80FD\u4ECE\u7B56\u6885\u6D1B-\u5F17\u862D\u514B\u723E\u96C6\u5408\u8AD6\uFF08ZF\uFF09\u7684\u516C\u7406\u63A8\u5BFC\u51FA\u6765\u3002\u53CD\u800C\u67D0\u4E9B\u9648\u8FF0\u7B49\u4EF7\u4E8E\u9009\u62E9\u516C\u7406\uFF08AC\uFF09\uFF0C\u800C\u5176\u4ED6\u7684\u5982\u5E03\u5C14\u7D20\u7406\u60F3\u5B9A\u7406\uFF0C\u4F53\u73B0\u4E86\u4E25\u683C\u5F31\u4E8EAC\u7684\u6027\u8D28\u3002\u7531\u4E8E\u8FD9\u4E2A\u5728ZF\u548CZF+AC (ZFC)\u4E4B\u95F4\u7684\u4E2D\u4ECB\u72B6\u6001\uFF0C\u5E03\u5C14\u7D20\u7406\u60F3\u5B9A\u7406\u7ECF\u5E38\u88AB\u63A5\u53D7\u4E3A\u96C6\u5408\u8BBA\u7684\u516C\u7406\u3002\u7ECF\u5E38\u7528\u7F29\u5199BPI\uFF08\u5BF9\u5E03\u5C14\u4EE3\u6570\uFF09\u6216PIT\u63D0\u53CA\u8FD9\u4E2A\u989D\u5916\u516C\u7406\u3002"@zh . . . "15804"^^ . . "1087342940"^^ . . . . "Boolean prime ideal theorem"@en . . . "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u043F\u0440\u043E \u0431\u0443\u043B\u0435\u0432\u0456 \u043F\u0440\u043E\u0441\u0442\u0456 \u0456\u0434\u0435\u0430\u043B\u0438"@uk . . . . . . . . . . . . "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u043F\u0440\u043E \u0411\u0443\u043B\u0435\u0432\u0456 \u043F\u0440\u043E\u0441\u0442\u0456 \u0456\u0434\u0435\u0430\u043B\u0438 \u0432 \u0442\u0435\u043E\u0440\u0456\u0457 \u043F\u043E\u0440\u044F\u0434\u043A\u0443 \u0441\u0442\u0432\u0435\u0440\u0434\u0436\u0443\u0454, \u0449\u043E \u0456\u0434\u0435\u0430\u043B\u0438 \u0432 \u0431\u0443\u043B\u0435\u0432\u0456\u0439 \u0430\u043B\u0433\u0435\u0431\u0440\u0456 \u043C\u043E\u0436\u0443\u0442\u044C \u0431\u0443\u0442\u0438 \u0440\u043E\u0437\u0448\u0438\u0440\u0435\u043D\u0456 \u0434\u043E \u043F\u0440\u043E\u0441\u0442\u0438\u0445 \u0456\u0434\u0435\u0430\u043B\u0456\u0432. \u0422\u0430\u043A \u044F\u043A \u0432 \u0442\u0435\u043E\u0440\u0456\u0457 \u043F\u043E\u0440\u044F\u0434\u043A\u0443 \u0431\u0456\u043B\u044C\u0448\u0456\u0441\u0442\u044C \u043F\u043E\u043D\u044F\u0442\u044C \u0454 , \u0456 \u0434\u0432\u043E\u0457\u0441\u0442\u0438\u043C \u0434\u043E \u0456\u0434\u0435\u0430\u043B\u0430 \u0454 \u0444\u0456\u043B\u044C\u0442\u0440, \u0442\u043E \u0430\u043D\u0430\u043B\u043E\u0433\u0456\u0447\u043D\u0435 \u0442\u0432\u0435\u0440\u0434\u0436\u0435\u043D\u043D\u044F \u0434\u043B\u044F \u0444\u0456\u043B\u044C\u0442\u0440\u0456\u0432 \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u2014 . \u0406\u0441\u043D\u0443\u044E\u0442\u044C \u0430\u043D\u0430\u043B\u043E\u0433\u0456\u0447\u043D\u0456 \u0444\u043E\u0440\u043C\u0443\u043B\u044E\u0432\u0430\u043D\u043D\u044F \u0456 \u0434\u043B\u044F \u0456\u043D\u0448\u0438\u0445 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0457\u0447\u043D\u0438\u0445 \u0441\u0442\u0440\u0443\u043A\u0442\u0443\u0440, \u043D\u0430\u043F\u0440\u0438\u043A\u043B\u0430\u0434, \u0434\u043B\u044F \u043A\u0456\u043B\u0435\u0446\u044C \u0442\u0430 \u0457\u0445 \u043F\u0440\u043E\u0441\u0442\u0438\u0445 \u0456\u0434\u0435\u0430\u043B\u0456\u0432 (\u0432 \u0442\u0435\u043E\u0440\u0456\u0457 \u043A\u0456\u043B\u0435\u0446\u044C). \u0412\u0441\u0456 \u0446\u0456 \u0444\u043E\u0440\u043C\u0443\u043B\u044E\u0432\u0430\u043D\u043D\u044F \u043D\u0435 \u043C\u043E\u0436\u0443\u0442\u044C \u0431\u0443\u0442\u0438 \u0434\u043E\u0432\u0435\u0434\u0435\u043D\u0456 \u0432 \u0440\u0430\u043C\u043A\u0430\u0445 \u0430\u043A\u0441\u0456\u043E\u043C \u0442\u0435\u043E\u0440\u0456\u0457 \u043C\u043D\u043E\u0436\u0438\u043D \u0426\u0435\u0440\u043C\u0435\u043B\u043E-\u0424\u0440\u0435\u043D\u0446\u0435\u043B\u044F (ZF). \u0412 \u0440\u0430\u043C\u043A\u0430\u0445 ZFC \u0434\u0435\u044F\u043A\u0456 \u0437 \u043D\u0438\u0445 \u0435\u043A\u0432\u0456\u0432\u0430\u043B\u0435\u043D\u0442\u043D\u0456 \u0430\u043A\u0441\u0456\u043E\u043C\u0456 \u0432\u0438\u0431\u043E\u0440\u0443 (AC), \u0430 \u0441\u0430\u043C\u0435 \u0442\u0435\u043E\u0440\u0435\u043C\u0430 \u043F\u0440\u043E \u0411\u0443\u043B\u0435\u0432\u0456 \u043F\u0440\u043E\u0441\u0442\u0456 \u0456\u0434\u0435\u0430\u043B\u0438 (BPI) \u2014 \u0454 \u043D\u0430\u0431\u0430\u0433\u0430\u0442\u043E \u0441\u043B\u0430\u0431\u0448\u043E\u044E \u0437\u0430 AC."@uk . . . . "\u0422\u0435\u043E\u0440\u0435\u043C\u0430 \u043F\u0440\u043E \u0411\u0443\u043B\u0435\u0432\u0456 \u043F\u0440\u043E\u0441\u0442\u0456 \u0456\u0434\u0435\u0430\u043B\u0438 \u0432 \u0442\u0435\u043E\u0440\u0456\u0457 \u043F\u043E\u0440\u044F\u0434\u043A\u0443 \u0441\u0442\u0432\u0435\u0440\u0434\u0436\u0443\u0454, \u0449\u043E \u0456\u0434\u0435\u0430\u043B\u0438 \u0432 \u0431\u0443\u043B\u0435\u0432\u0456\u0439 \u0430\u043B\u0433\u0435\u0431\u0440\u0456 \u043C\u043E\u0436\u0443\u0442\u044C \u0431\u0443\u0442\u0438 \u0440\u043E\u0437\u0448\u0438\u0440\u0435\u043D\u0456 \u0434\u043E \u043F\u0440\u043E\u0441\u0442\u0438\u0445 \u0456\u0434\u0435\u0430\u043B\u0456\u0432. \u0422\u0430\u043A \u044F\u043A \u0432 \u0442\u0435\u043E\u0440\u0456\u0457 \u043F\u043E\u0440\u044F\u0434\u043A\u0443 \u0431\u0456\u043B\u044C\u0448\u0456\u0441\u0442\u044C \u043F\u043E\u043D\u044F\u0442\u044C \u0454 , \u0456 \u0434\u0432\u043E\u0457\u0441\u0442\u0438\u043C \u0434\u043E \u0456\u0434\u0435\u0430\u043B\u0430 \u0454 \u0444\u0456\u043B\u044C\u0442\u0440, \u0442\u043E \u0430\u043D\u0430\u043B\u043E\u0433\u0456\u0447\u043D\u0435 \u0442\u0432\u0435\u0440\u0434\u0436\u0435\u043D\u043D\u044F \u0434\u043B\u044F \u0444\u0456\u043B\u044C\u0442\u0440\u0456\u0432 \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u2014 . \u0406\u0441\u043D\u0443\u044E\u0442\u044C \u0430\u043D\u0430\u043B\u043E\u0433\u0456\u0447\u043D\u0456 \u0444\u043E\u0440\u043C\u0443\u043B\u044E\u0432\u0430\u043D\u043D\u044F \u0456 \u0434\u043B\u044F \u0456\u043D\u0448\u0438\u0445 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0457\u0447\u043D\u0438\u0445 \u0441\u0442\u0440\u0443\u043A\u0442\u0443\u0440, \u043D\u0430\u043F\u0440\u0438\u043A\u043B\u0430\u0434, \u0434\u043B\u044F \u043A\u0456\u043B\u0435\u0446\u044C \u0442\u0430 \u0457\u0445 \u043F\u0440\u043E\u0441\u0442\u0438\u0445 \u0456\u0434\u0435\u0430\u043B\u0456\u0432 (\u0432 \u0442\u0435\u043E\u0440\u0456\u0457 \u043A\u0456\u043B\u0435\u0446\u044C). \u0412\u0441\u0456 \u0446\u0456 \u0444\u043E\u0440\u043C\u0443\u043B\u044E\u0432\u0430\u043D\u043D\u044F \u043D\u0435 \u043C\u043E\u0436\u0443\u0442\u044C \u0431\u0443\u0442\u0438 \u0434\u043E\u0432\u0435\u0434\u0435\u043D\u0456 \u0432 \u0440\u0430\u043C\u043A\u0430\u0445 \u0430\u043A\u0441\u0456\u043E\u043C \u0442\u0435\u043E\u0440\u0456\u0457 \u043C\u043D\u043E\u0436\u0438\u043D \u0426\u0435\u0440\u043C\u0435\u043B\u043E-\u0424\u0440\u0435\u043D\u0446\u0435\u043B\u044F (ZF)."@uk . . . . . . "Twierdzenie o ideale pierwszym"@pl . . . "En math\u00E9matiques, un th\u00E9or\u00E8me de l'id\u00E9al premier garantit l'existence de certains types de sous-ensembles dans une alg\u00E8bre. Un exemple courant est le th\u00E9or\u00E8me de l'id\u00E9al premier dans une alg\u00E8bre de Boole, qui \u00E9nonce que tout id\u00E9al d'une alg\u00E8bre de Boole est inclus dans un id\u00E9al premier. Une variante de cet \u00E9nonc\u00E9 pour filtres sur des ensembles est connue comme le . D'autres th\u00E9or\u00E8mes sont obtenus en consid\u00E9rant les diff\u00E9rentes structures math\u00E9matiques avec les notions d'id\u00E9al appropri\u00E9es, par exemple, les anneaux et leurs id\u00E9aux premiers (en th\u00E9orie des anneaux), ou les treillis distributifs et leurs id\u00E9aux maximaux (en th\u00E9orie des ordres). Cet article se concentre sur le th\u00E9or\u00E8me de l'id\u00E9al premier en th\u00E9orie des ordres. Bien que les divers th\u00E9or\u00E8mes de l'id\u00E9al premier puissent para\u00EEtre simples et intuitifs, ils ne peuvent pas \u00EAtre d\u00E9duits en g\u00E9n\u00E9ral des axiomes de la th\u00E9orie des ensembles de Zermelo-Fraenkel sans l'axiome du choix (en abr\u00E9g\u00E9 ZF). Au lieu de cela, certains \u00E9nonc\u00E9s s'av\u00E8rent \u00E9quivalents \u00E0 l'axiome du choix (AC), tandis que d'autres \u2014 le th\u00E9or\u00E8me de l'id\u00E9al premier dans une alg\u00E8bre de Boole, par exemple \u2014 constituent une propri\u00E9t\u00E9 strictement plus faible que AC. C'est gr\u00E2ce \u00E0 ce statut interm\u00E9diaire entre ZF et ZF + AC (ZFC) que le th\u00E9or\u00E8me de l'id\u00E9al premier dans une alg\u00E8bre de Boole est souvent pris comme un axiome de la th\u00E9orie des ensembles. Les abr\u00E9viations BPI ou PIT (pour les alg\u00E8bres de Boole) sont parfois utilis\u00E9es pour se r\u00E9f\u00E9rer \u00E0 cet axiome suppl\u00E9mentaire."@fr . . . . "In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by considering different mathematical structures with appropriate notions of ideals, for example, rings and prime ideals (of ring theory), or distributive lattices and maximal ideals (of order theory). This article focuses on prime ideal theorems from order theory."@en . "314919"^^ . . . . "Teorema do ideal primo booliano"@pt . . . . . . "November 2021"@en . . . . . "Twierdzenie o ideale pierwszym \u2013 twierdzenie teorii krat rozdzielnych."@pl . . "Twierdzenie o ideale pierwszym \u2013 twierdzenie teorii krat rozdzielnych."@pl . . . "Der boolesche Primidealsatz sagt aus, dass jede boolesche Algebra ein Primideal enth\u00E4lt. Der Beweis dieses Satzes kann nicht ohne transfinite Methoden gef\u00FChrt werden, das bedeutet, dass er nicht aus den Axiomen der Mengenlehre ohne Auswahlaxiom beweisbar ist. Umgekehrt ist das Auswahlaxiom nicht aus dem booleschen Primidealsatz beweisbar, dieser Satz ist also schw\u00E4cher als das Auswahlaxiom. Au\u00DFerdem ist der Satz (relativ zu den Axiomen der Zermelo-Fraenkel-Mengenlehre) \u00E4quivalent zu einigen anderen S\u00E4tzen wie zum Beispiel G\u00F6dels Vollst\u00E4ndigkeitssatz. (Das bedeutet, dass man aus den Axiomen der Mengenlehre plus dem booleschen Primidealsatz dieselben S\u00E4tze beweisen kann wie aus den Axiomen der Mengenlehre plus dem g\u00F6delschen Vollst\u00E4ndigkeitssatz.)"@de . . . "Th\u00E9or\u00E8me de l'id\u00E9al premier dans une alg\u00E8bre de Boole"@fr . . . "Does this specific point of view match the title?"@en . . . "Em matem\u00E1tica, um teorema do ideal primo garante a exist\u00EAncia de certos tipos de subconjuntos numa \u00E1lgebra dada. Um exemplo comum \u00E9 o teorema do ideal primo booleano, o qual afirma que ideais em uma \u00E1lgebra booleana podem ser estendidos para ideais primos. Uma varia\u00E7\u00E3o dessa afirma\u00E7\u00E3o para filtros em conjuntos \u00E9 conhecida como o . Outros teoremas s\u00E3o obtidos considerando diferentes estruturas matem\u00E1ticas com no\u00E7\u00F5es apropriadas de ideais, por exemplo, an\u00E9is e ideais primos (da teoria dos an\u00E9is), ou e ideais maximais (de ). Esse artigo foca nos teoremas do ideal primo da ."@pt . . . "\u7D20\u7406\u60F3\u5B9A\u7406\uFF08prime ideal theorem\uFF09\u5373\u4FDD\u8BC1\u5728\u7ED9\u5B9A\u7684\u62BD\u8C61\u4EE3\u6570\u4E2D\u7279\u5B9A\u7C7B\u578B\u4E4B\u5B50\u96C6\u7684\u5B58\u5728\u6027\u4E4B\u6578\u5B78\u5B9A\u7406\u3002\u5E38\u89C1\u7684\u4F8B\u5B50\u5C31\u662F\u5E03\u5C14\u7D20\u7406\u60F3\u5B9A\u7406\uFF08Boolean prime ideal theorem\uFF09\uFF0C\u5B83\u58F0\u79F0\u5728\u5E03\u5C14\u4EE3\u6570\u4E2D\u7684\u7406\u60F3\u53EF\u4EE5\u88AB\u6269\u5C55\u6210\u7D20\u7406\u60F3\u3002\u8FD9\u4E2A\u9648\u8FF0\u5BF9\u4E8E\u5728\u96C6\u5408\u4E0A\u7684\u6EE4\u5B50\u7684\u53D8\u4F53\u53EB\u505A\u53EB\u505A\u3002\u901A\u8FC7\u8003\u8651\u4E0D\u540C\u7684\u5E26\u6709\u9002\u5F53\u7684\u7406\u60F3\u6982\u5FF5\u7684\u6570\u5B66\u7ED3\u6784\u53EF\u83B7\u5F97\u5176\u4ED6\u5B9A\u7406\uFF0C\u4F8B\u5982\u74B0\u548C\uFF08\u73AF\u8BBA\u7684\uFF09\u7D20\u7406\u60F3\uFF0C\u548C\u5206\u914D\u683C\u548C\uFF08\u5E8F\u7406\u8BBA\u7684\uFF09\u7684\u6781\u5927\u7406\u60F3\u3002\u672C\u6587\u5173\u6CE8\u5E8F\u7406\u8BBA\u7684\u7D20\u7406\u60F3\u5B9A\u7406\u3002 \u5C3D\u7BA1\u5404\u79CD\u7D20\u7406\u60F3\u5B9A\u7406\u53EF\u80FD\u770B\u8D77\u6765\u7B80\u5355\u4E14\u76F4\u89C9\uFF0C\u5B83\u4EEC\u4E00\u822C\u4E0D\u80FD\u4ECE\u7B56\u6885\u6D1B-\u5F17\u862D\u514B\u723E\u96C6\u5408\u8AD6\uFF08ZF\uFF09\u7684\u516C\u7406\u63A8\u5BFC\u51FA\u6765\u3002\u53CD\u800C\u67D0\u4E9B\u9648\u8FF0\u7B49\u4EF7\u4E8E\u9009\u62E9\u516C\u7406\uFF08AC\uFF09\uFF0C\u800C\u5176\u4ED6\u7684\u5982\u5E03\u5C14\u7D20\u7406\u60F3\u5B9A\u7406\uFF0C\u4F53\u73B0\u4E86\u4E25\u683C\u5F31\u4E8EAC\u7684\u6027\u8D28\u3002\u7531\u4E8E\u8FD9\u4E2A\u5728ZF\u548CZF+AC (ZFC)\u4E4B\u95F4\u7684\u4E2D\u4ECB\u72B6\u6001\uFF0C\u5E03\u5C14\u7D20\u7406\u60F3\u5B9A\u7406\u7ECF\u5E38\u88AB\u63A5\u53D7\u4E3A\u96C6\u5408\u8BBA\u7684\u516C\u7406\u3002\u7ECF\u5E38\u7528\u7F29\u5199BPI\uFF08\u5BF9\u5E03\u5C14\u4EE3\u6570\uFF09\u6216PIT\u63D0\u53CA\u8FD9\u4E2A\u989D\u5916\u516C\u7406\u3002"@zh . . . . . . . "En math\u00E9matiques, un th\u00E9or\u00E8me de l'id\u00E9al premier garantit l'existence de certains types de sous-ensembles dans une alg\u00E8bre. Un exemple courant est le th\u00E9or\u00E8me de l'id\u00E9al premier dans une alg\u00E8bre de Boole, qui \u00E9nonce que tout id\u00E9al d'une alg\u00E8bre de Boole est inclus dans un id\u00E9al premier. Une variante de cet \u00E9nonc\u00E9 pour filtres sur des ensembles est connue comme le . D'autres th\u00E9or\u00E8mes sont obtenus en consid\u00E9rant les diff\u00E9rentes structures math\u00E9matiques avec les notions d'id\u00E9al appropri\u00E9es, par exemple, les anneaux et leurs id\u00E9aux premiers (en th\u00E9orie des anneaux), ou les treillis distributifs et leurs id\u00E9aux maximaux (en th\u00E9orie des ordres). Cet article se concentre sur le th\u00E9or\u00E8me de l'id\u00E9al premier en th\u00E9orie des ordres."@fr . . "In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by considering different mathematical structures with appropriate notions of ideals, for example, rings and prime ideals (of ring theory), or distributive lattices and maximal ideals (of order theory). This article focuses on prime ideal theorems from order theory. Although the various prime ideal theorems may appear simple and intuitive, they cannot be deduced in general from the axioms of Zermelo\u2013Fraenkel set theory without the axiom of choice (abbreviated ZF). Instead, some of the statements turn out to be equivalent to the axiom of choice (AC), while others\u2014the Boolean prime ideal theorem, for instance\u2014represent a property that is strictly weaker than AC. It is due to this intermediate status between ZF and ZF + AC (ZFC) that the Boolean prime ideal theorem is often taken as an axiom of set theory. The abbreviations BPI or PIT (for Boolean algebras) are sometimes used to refer to this additional axiom."@en . . . . . . "\u5E03\u5C14\u7D20\u7406\u60F3\u5B9A\u7406"@zh . . . . . "Em matem\u00E1tica, um teorema do ideal primo garante a exist\u00EAncia de certos tipos de subconjuntos numa \u00E1lgebra dada. Um exemplo comum \u00E9 o teorema do ideal primo booleano, o qual afirma que ideais em uma \u00E1lgebra booleana podem ser estendidos para ideais primos. Uma varia\u00E7\u00E3o dessa afirma\u00E7\u00E3o para filtros em conjuntos \u00E9 conhecida como o . Outros teoremas s\u00E3o obtidos considerando diferentes estruturas matem\u00E1ticas com no\u00E7\u00F5es apropriadas de ideais, por exemplo, an\u00E9is e ideais primos (da teoria dos an\u00E9is), ou e ideais maximais (de ). Esse artigo foca nos teoremas do ideal primo da . Embora os v\u00E1rios teoremas do ideal primo possam parecer simples e intuitivos, eles geralmente n\u00E3o podem ser derivados dos axiomas da teoria dos conjuntos de Zermelo-Fraenkel sem o axioma da escolha (abreviado ZF). Em vez disso, algumas das afirma\u00E7\u00F5es acabam sendo equivalentes ao axioma da escolha (AC = Axiom of choice), enquanto outros \u2013 o teorema do ideal primo booleano, por exemplo - representam uma propriedade que \u00E9 estritamente mais fraca que AC. Devido a este estado intermedi\u00E1rio entre ZF e ZF + AC (ZFC) que o teorema do ideal primo booleano \u00E9 frequentemente considerado um axioma da teoria dos conjuntos. As abrevia\u00E7\u00F5es BPI (Boolean Prime Ideal, em portugu\u00EAs ideal primo booleano, IPB) ou PIT (Prime Ideal Teorem, teorema do ideal primo em portugu\u00EAs, TIP) (para \u00E1lgebras booleanas) s\u00E3o por vezes usadas para se referir a esse axioma adicional."@pt . . "Boolescher Primidealsatz"@de . . "Der boolesche Primidealsatz sagt aus, dass jede boolesche Algebra ein Primideal enth\u00E4lt. Der Beweis dieses Satzes kann nicht ohne transfinite Methoden gef\u00FChrt werden, das bedeutet, dass er nicht aus den Axiomen der Mengenlehre ohne Auswahlaxiom beweisbar ist. Umgekehrt ist das Auswahlaxiom nicht aus dem booleschen Primidealsatz beweisbar, dieser Satz ist also schw\u00E4cher als das Auswahlaxiom. Au\u00DFerdem ist der Satz (relativ zu den Axiomen der Zermelo-Fraenkel-Mengenlehre) \u00E4quivalent zu einigen anderen S\u00E4tzen wie zum Beispiel G\u00F6dels Vollst\u00E4ndigkeitssatz. (Das bedeutet, dass man aus den Axiomen der Mengenlehre plus dem booleschen Primidealsatz dieselben S\u00E4tze beweisen kann wie aus den Axiomen der Mengenlehre plus dem g\u00F6delschen Vollst\u00E4ndigkeitssatz.) Ersetzt man die boolesche Algebra durch ihre duale boolesche Algebra, so wird der boolesche Primidealsatz zum Ultrafilterlemma."@de . . . . . . . . . .