[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki5\/archives\/6944#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki5\/archives\/6944","headline":"\u4e3b\u30a4\u30c7\u30a2\u30eb\u306b\u95a2\u3059\u308b\u6607\u9396\u6761\u4ef6 – Wikipedia","name":"\u4e3b\u30a4\u30c7\u30a2\u30eb\u306b\u95a2\u3059\u308b\u6607\u9396\u6761\u4ef6 – Wikipedia","description":"\u62bd\u8c61\u4ee3\u6570\u5b66\u306b\u304a\u3044\u3066\u3001\u6607\u9396\u6761\u4ef6\u306f\u5305\u542b\u95a2\u4fc2\u306b\u3088\u308b\u534a\u9806\u5e8f\u304c\u5165\u3063\u305f\u74b0\u306e\u4e3b\u5de6\u3001\u4e3b\u53f3\u3001\u3042\u308b\u3044\u306f\u4e3b\u4e21\u5074\u30a4\u30c7\u30a2\u30eb\u306e\u534a\u9806\u5e8f\u96c6\u5408\u306b\u9069\u7528\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u4e3b\u30a4\u30c7\u30a2\u30eb\u306b\u95a2\u3059\u308b\u6607\u9396\u6761\u4ef6 (ascending chain condition on principal ideals) \uff08ACCP \u3068\u7701\u7565\u3055\u308c\u308b\uff09\u304c\u6e80\u305f\u3055\u308c\u308b\u3068\u306f\u3001\u74b0\u306b\u304a\u3044\u3066\u4e0e\u3048\u3089\u308c\u305f\u30bf\u30a4\u30d7\uff08\u5de6\uff0f\u53f3\uff0f\u4e21\u5074\uff09\u306e\u4e3b\u30a4\u30c7\u30a2\u30eb\u306e\u771f\u306e\u7121\u9650\u6607\u9396\u304c\u5b58\u5728\u3057\u306a\u3044\u3068\u3044\u3046\u3053\u3068\u3067\u3042\u308b\u3002\u3042\u308b\u3044\u306f\u5225\u306e\u8a00\u3044\u65b9\u3092\u3059\u308c\u3070\u3001\u3059\u3079\u3066\u306e\u6607\u9396\u306f\u3084\u304c\u3066\u4e00\u5b9a\u306b\u306a\u308b\u3002 \u7247\u5272\u308c\u3067\u3042\u308b\u964d\u9396\u6761\u4ef6\u3082\u307e\u305f\u3053\u308c\u3089\u306e\u534a\u9806\u5e8f\u96c6\u5408\u306b\u9069\u7528\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u304c\u3001\u3057\u304b\u3057\u7528\u8a9e “DCCP” \u306e\u5fc5\u8981\u306f\u73fe\u5728\u306f\u5168\u304f\u306a\u3044\u3001\u306a\u305c\u306a\u3089\u3070\u305d\u306e\u3088\u3046\u306a\u74b0\u306f\u65e2\u306b\u5de6\u3042\u308b\u3044\u306f\u53f3\u5b8c\u5168\u74b0\u3068\u3044\u3046\u540d\u524d\u304c\u3064\u3044\u3066\u3044\u308b\u304b\u3089\u3067\u3042\u308b\u3002\uff08\u4e0b\u306e\u975e\u53ef\u63db\u74b0\u306e\u7bc0\u3092\u53c2\u7167\u3002\uff09 \u30cd\u30fc\u30bf\u30fc\u74b0\uff08\u4f8b\u3048\u3070\u4e3b\u30a4\u30c7\u30a2\u30eb\u6574\u57df\uff09\u306f\u5178\u578b\u7684\u306a\u4f8b\u3067\u3042\u308b\u304c\u3001\u3044\u304f\u3064\u304b\u306e\u91cd\u8981\u306a\u975e\u30cd\u30fc\u30bf\u30fc\u74b0\u3001\u7279\u306b\u4e00\u610f\u5206\u89e3\u6574\u57df\u3068\u5de6\u307e\u305f\u306f\u53f3\u5b8c\u5168\u74b0\u3082\u307e\u305f (ACCP) \u3092\u6e80\u305f\u3059\u3002 \u30cd\u30fc\u30bf\u30fc\u6574\u57df\u306b\u304a\u3044\u3066 0 \u3067\u306a\u3044\u975e\u5358\u5143\u306f\u65e2\u7d04\u5143\u306b\u5206\u89e3\u3059\u308b\u3068\u3044\u3046\u3053\u3068\u306f\u3088\u304f\u77e5\u3089\u308c\u3066\u3044\u308b\u3002\u3053\u306e\u3053\u3068\u306e\u8a3c\u660e\u306f (ACC) \u3067\u306f\u306a\u304f","datePublished":"2017-11-29","dateModified":"2017-11-29","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki5\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki5\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5de3d18759e109d4aecac8ecd7376befbccda5bc","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5de3d18759e109d4aecac8ecd7376befbccda5bc","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/jp\/wiki5\/archives\/6944","about":["Wiki"],"wordCount":2242,"articleBody":"\u62bd\u8c61\u4ee3\u6570\u5b66\u306b\u304a\u3044\u3066\u3001\u6607\u9396\u6761\u4ef6\u306f\u5305\u542b\u95a2\u4fc2\u306b\u3088\u308b\u534a\u9806\u5e8f\u304c\u5165\u3063\u305f\u74b0\u306e\u4e3b\u5de6\u3001\u4e3b\u53f3\u3001\u3042\u308b\u3044\u306f\u4e3b\u4e21\u5074\u30a4\u30c7\u30a2\u30eb\u306e\u534a\u9806\u5e8f\u96c6\u5408\u306b\u9069\u7528\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u4e3b\u30a4\u30c7\u30a2\u30eb\u306b\u95a2\u3059\u308b\u6607\u9396\u6761\u4ef6 (ascending chain condition on principal ideals) \uff08ACCP \u3068\u7701\u7565\u3055\u308c\u308b\uff09\u304c\u6e80\u305f\u3055\u308c\u308b\u3068\u306f\u3001\u74b0\u306b\u304a\u3044\u3066\u4e0e\u3048\u3089\u308c\u305f\u30bf\u30a4\u30d7\uff08\u5de6\uff0f\u53f3\uff0f\u4e21\u5074\uff09\u306e\u4e3b\u30a4\u30c7\u30a2\u30eb\u306e\u771f\u306e\u7121\u9650\u6607\u9396\u304c\u5b58\u5728\u3057\u306a\u3044\u3068\u3044\u3046\u3053\u3068\u3067\u3042\u308b\u3002\u3042\u308b\u3044\u306f\u5225\u306e\u8a00\u3044\u65b9\u3092\u3059\u308c\u3070\u3001\u3059\u3079\u3066\u306e\u6607\u9396\u306f\u3084\u304c\u3066\u4e00\u5b9a\u306b\u306a\u308b\u3002  \u7247\u5272\u308c\u3067\u3042\u308b\u964d\u9396\u6761\u4ef6\u3082\u307e\u305f\u3053\u308c\u3089\u306e\u534a\u9806\u5e8f\u96c6\u5408\u306b\u9069\u7528\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u304c\u3001\u3057\u304b\u3057\u7528\u8a9e “DCCP” \u306e\u5fc5\u8981\u306f\u73fe\u5728\u306f\u5168\u304f\u306a\u3044\u3001\u306a\u305c\u306a\u3089\u3070\u305d\u306e\u3088\u3046\u306a\u74b0\u306f\u65e2\u306b\u5de6\u3042\u308b\u3044\u306f\u53f3\u5b8c\u5168\u74b0\u3068\u3044\u3046\u540d\u524d\u304c\u3064\u3044\u3066\u3044\u308b\u304b\u3089\u3067\u3042\u308b\u3002\uff08\u4e0b\u306e\u975e\u53ef\u63db\u74b0\u306e\u7bc0\u3092\u53c2\u7167\u3002\uff09\u30cd\u30fc\u30bf\u30fc\u74b0\uff08\u4f8b\u3048\u3070\u4e3b\u30a4\u30c7\u30a2\u30eb\u6574\u57df\uff09\u306f\u5178\u578b\u7684\u306a\u4f8b\u3067\u3042\u308b\u304c\u3001\u3044\u304f\u3064\u304b\u306e\u91cd\u8981\u306a\u975e\u30cd\u30fc\u30bf\u30fc\u74b0\u3001\u7279\u306b\u4e00\u610f\u5206\u89e3\u6574\u57df\u3068\u5de6\u307e\u305f\u306f\u53f3\u5b8c\u5168\u74b0\u3082\u307e\u305f (ACCP) \u3092\u6e80\u305f\u3059\u3002\u30cd\u30fc\u30bf\u30fc\u6574\u57df\u306b\u304a\u3044\u3066 0 \u3067\u306a\u3044\u975e\u5358\u5143\u306f\u65e2\u7d04\u5143\u306b\u5206\u89e3\u3059\u308b\u3068\u3044\u3046\u3053\u3068\u306f\u3088\u304f\u77e5\u3089\u308c\u3066\u3044\u308b\u3002\u3053\u306e\u3053\u3068\u306e\u8a3c\u660e\u306f (ACC) \u3067\u306f\u306a\u304f (ACCP) \u306e\u307f\u306b\u983c\u3063\u3066\u3044\u308b\u306e\u3067\u3001(ACCP) \u306e\u6210\u308a\u7acb\u3064\u4efb\u610f\u306e\u6574\u57df\u306b\u304a\u3044\u3066\u3001\u65e2\u7d04\u5143\u5206\u89e3\u304c\u5b58\u5728\u3059\u308b\u3002\uff08\u8a00\u3044\u63db\u3048\u308b\u3068\u3001(ACCP) \u306e\u6210\u308a\u7acb\u3064\u4efb\u610f\u306e\u6574\u57df\u306f\u539f\u5b50\u6574\u57df\uff08\u82f1\u8a9e\u7248\uff09\u3067\u3042\u308b\u3002\u3057\u304b\u3057\u9006\u306f\u3001(Grams 1974) \u306b\u304a\u3044\u3066\u8a3c\u660e\u3055\u308c\u3066\u3044\u308b\u3088\u3046\u306b\u3001\u9593\u9055\u3044\u3067\u3042\u308b\u3002\uff09\u305d\u306e\u3088\u3046\u306a\u5206\u89e3\u306f\u4e00\u610f\u3067\u306a\u3044\u304b\u3082\u3057\u308c\u306a\u3044\u3002\u5206\u89e3\u306e\u4e00\u610f\u6027\u3092\u8a3c\u660e\u3059\u308b\u901a\u5e38\u306e\u65b9\u6cd5\u306f\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u88dc\u984c\u3092\u4f7f\u3046\u304c\u3001\u3053\u308c\u306f\u56e0\u5b50\u304c\u5358\u306b\u65e2\u7d04\u3067\u3042\u308b\u3060\u3051\u3067\u306a\u304f\u7d20\u5143\u3067\u3042\u308b\u3053\u3068\u3092\u8981\u6c42\u3059\u308b\u3002\u5b9f\u969b\u3001\u6b21\u306e\u7279\u5fb4\u3065\u3051\u304c\u3042\u308b\uff1a A \u3092\u6574\u57df\u3068\u3059\u308b\u3002\u3053\u306e\u3068\u304d\u4ee5\u4e0b\u306f\u540c\u5024\u3067\u3042\u308b\u3002  A \u306f UFD \u3067\u3042\u308b\u3002A \u306f (ACCP) \u3092\u6e80\u305f\u3057\u3001A \u306e\u3059\u3079\u3066\u306e\u65e2\u7d04\u5143\u306f\u7d20\u5143\u3067\u3042\u308b\u3002A \u306f (ACCP) \u3092\u6e80\u305f\u3059GCD\u6574\u57df\u3067\u3042\u308b\u3002\u3044\u308f\u3086\u308b\u6c38\u7530\u5224\u5b9a\u6cd5 (Nagata criterion) \u304c (ACCP) \u3092\u6e80\u305f\u3059\u6574\u57df A \u306b\u5bfe\u3057\u3066\u6210\u308a\u7acb\u3064\uff1a S \u3092\u7d20\u5143\u3067\u751f\u6210\u3055\u308c\u308b A \u306e\u4e57\u6cd5\u7684\u9589\u90e8\u5206\u96c6\u5408\u3068\u3059\u308b\u3002\u5c40\u6240\u5316 S\u22121A \u304c UFD \u3067\u3042\u308c\u3070\u3001A \u3082 UFD \u3067\u3042\u308b\u3002(Nagata & 1975, Lemma 2.1) \uff08\u3053\u308c\u306e\u9006\u306f\u81ea\u660e\u3067\u3042\u308b\u3053\u3068\u3092\u6ce8\u610f\u3057\u3088\u3046\u3002\uff09\u6574\u57df A \u304c (ACCP) \u3092\u6e80\u305f\u3059\u3053\u3068\u3068\u591a\u9805\u5f0f\u74b0 A[t] \u304c (ACCP) \u3092\u6e80\u305f\u3059\u3053\u3068\u306f\u540c\u5024\u3067\u3042\u308b[1]\u3002A \u304c\u6574\u57df\u3067\u306a\u3044\u3068\u304d\u985e\u4f3c\u306e\u4e3b\u5f35\u306f\u8aa4\u308a\u3067\u3042\u308b\u3002\u3059\u3079\u3066\u306e\u6709\u9650\u751f\u6210\u30a4\u30c7\u30a2\u30eb\u304c\u4e3b\u3067\u3042\u308b\u3088\u3046\u306a\u6574\u57df\uff08\u3059\u306a\u308f\u3061\u30d9\u30ba\u30fc\u6574\u57df\uff09\u304c (ACCP) \u3092\u6e80\u305f\u3059\u3053\u3068\u3068\u305d\u308c\u304c\u4e3b\u30a4\u30c7\u30a2\u30eb\u6574\u57df\u3067\u3042\u308b\u3053\u3068\u306f\u540c\u5024\u3067\u3042\u308b[3]\u3002\u5b9a\u6570\u9805\u304c\u6574\u6570\u3067\u3042\u308b\u3059\u3079\u3066\u306e\u6709\u7406\u4fc2\u6570\u591a\u9805\u5f0f\u304b\u3089\u306a\u308b\u74b0 Z+XQ[X] \u306f (ACCP) \u3092\u6e80\u305f\u3055\u306a\u3044\u6574\u57df\uff08\u5b9f\u306f GCD \u6574\u57df\uff09\u306e\u4f8b\u3067\u3042\u308b\u3001\u3068\u3044\u3046\u306e\u3082\u4e3b\u30a4\u30c7\u30a2\u30eb\u306e\u9396  (X)\u2282(X\/2)\u2282(X\/4)\u2282(X\/8),...{displaystyle (X)subset (X\/2)subset (X\/4)subset (X\/8),…}\u306f\u7121\u9650\u306b\u7d9a\u304f\u304b\u3089\u3067\u3042\u308b\u3002\u975e\u53ef\u63db\u74b0[\u7de8\u96c6]\u975e\u53ef\u63db\u306e\u5834\u5408\u306b\u306f\u3001\u53f3 ACCP \u3068\u5de6 ACCP \u3092\u533a\u5225\u3059\u308b\u5fc5\u8981\u304c\u51fa\u3066\u304f\u308b\u3002\u524d\u8005\u306f xR \u306e\u5f62\u306e\u30a4\u30c7\u30a2\u30eb\u306e\u534a\u9806\u5e8f\u96c6\u5408\u304c\u6607\u9396\u6761\u4ef6\u3092\u6e80\u305f\u3059\u3068\u3044\u3046\u3053\u3068\u3092\u8981\u6c42\u3059\u308b\u3060\u3051\u3067\u3042\u308a\u3001\u5f8c\u8005\u306f Rx \u306e\u5f62\u306e\u30a4\u30c7\u30a2\u30eb\u306e\u534a\u9806\u5e8f\u96c6\u5408\u3092\u691c\u67fb\u3059\u308b\u3060\u3051\u3067\u3042\u308b\u3002\u4eca\u306f “Bass’ Theorem P” \u3068\u547c\u3070\u308c\u3066\u3044\u308b\u3001(Bass 1960) \u306b\u3042\u308b Hyman Bass\uff08\u82f1\u8a9e\u7248\uff09 \u306b\u3088\u308b\u5b9a\u7406\u306f\u3001\u74b0 R \u306e\u4e3b\u5de6\u30a4\u30c7\u30a2\u30eb\u306b\u3064\u3044\u3066\u306e\u964d\u9396\u6761\u4ef6\u306f R \u304c\u53f3\u5b8c\u5168\u74b0\u3067\u3042\u308b\u3053\u3068\u3068\u540c\u5024\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3057\u305f\u3002D. Jonah \u306f (Jonah 1970) \u306b\u304a\u3044\u3066 ACCP \u3068\u5b8c\u5168\u74b0\u306e\u9593\u306b side-switching connection \u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u3092\u793a\u3057\u305f\u3002R \u304c\u53f3\u5b8c\u5168\uff08\u53f3 DCCP \u3092\u6e80\u305f\u3059\uff09\u306a\u3089\u3070 R \u306f\u5de6 ACCP \u3092\u6e80\u305f\u3059\u3053\u3068\u3068\u3001\u5bfe\u79f0\u7684\u306b\u3001R \u304c\u5de6\u5b8c\u5168\uff08\u5de6 DCCP \u3092\u6e80\u305f\u3059\uff09\u306a\u3089\u3070\u53f3 ACCP \u3092\u6e80\u305f\u3059\u3053\u3068\u304c\u793a\u3055\u308c\u305f\u3002\u9006\u306f\u6b63\u3057\u304f\u306a\u304f\u3001\u4e0a\u306e\u5de6\u3068\u53f3\u306e\u5207\u308a\u66ff\u3048\u306f\u6253\u3061\u9593\u9055\u3044\u3067\u306f\u306a\u3044\u3002ACCP \u304c R \u306e\u53f3\u5074\u306b\u3064\u3044\u3066\u6210\u308a\u7acb\u3068\u3046\u3068\u5de6\u5074\u306b\u3064\u3044\u3066\u6210\u308a\u7acb\u3068\u3046\u3068\u3001\u305d\u308c\u306f R \u304c 0 \u3067\u306a\u3044\u76f4\u4ea4\u51aa\u7b49\u5143\u306e\u7121\u9650\u96c6\u5408\u3092\u6301\u305f\u306a\u3044\u3053\u3068\u3068 R \u304c\u30c7\u30c7\u30ad\u30f3\u30c8\u6709\u9650\u74b0\u3067\u3042\u308b\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002^ Gilmer, Robert (1986), \u201cProperty E in commutative monoid rings\u201d, Group and semigroup rings (Johannesburg, 1985), North-Holland Math. Stud., 126, Amsterdam: North-Holland, pp.\u00a013\u201318, MR860048, http:\/\/books.google.com\/books?id=Ed3FiiVkKcsC&pg=PA15\u00a0.^ \u8a3c\u660e\uff1a \u30d9\u30ba\u30fc\u6574\u57df\u306b\u304a\u3044\u3066 ACCP \u306f\u6709\u9650\u751f\u6210\u30a4\u30c7\u30a2\u30eb\u306b\u95a2\u3059\u308b ACC \u306b\u540c\u5024\u3067\u3042\u308b\u304c\u3001\u3053\u308c\u306f\u3059\u3079\u3066\u306e\u30a4\u30c7\u30a2\u30eb\u306b\u95a2\u3059\u308b ACC \u306b\u540c\u5024\u3067\u3042\u308b\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u308b\u3002\u3057\u305f\u304c\u3063\u3066\u305d\u306e\u6574\u57df\u306f\u30cd\u30fc\u30bf\u30fc\u304b\u3064\u30d9\u30ba\u30fc\u3067\u3042\u308a\u3001\u3086\u3048\u306b\u4e3b\u30a4\u30c7\u30a2\u30eb\u6574\u57df\u3067\u3042\u308b\u3002\u53c2\u8003\u6587\u732e[\u7de8\u96c6]Bass, Hyman (1960), \u201cFinitistic dimension and a homological generalization of semi-primary rings\u201d, Trans. Amer. Math. Soc. 95:  466\u2013488, doi:10.1090\/s0002-9947-1960-0157984-8, ISSN\u00a00002-9947, MR0157984\u00a0Grams, Anne (1974), \u201cAtomic rings and the ascending chain condition for principal ideals\u201d, Proc. Cambridge Philos. Soc. 75:  321\u2013329, doi:10.1017\/s0305004100048532, MR0340249\u00a0Heinzer, William J.; Lantz, David C. (1994), \u201cACCP in polynomial rings: a counterexample\u201d, Proc. Amer. Math. Soc. 121 (3):  975\u2013977, doi:10.2307\/2160301, ISSN\u00a00002-9939, JSTOR\u00a02160301, MR1232140, https:\/\/jstor.org\/stable\/2160301\u00a0Jonah, David (1970), \u201cRings with the minimum condition for principal right ideals have the maximum condition for principal left ideals\u201d, Math. Z. 113:  106\u2013112, doi:10.1007\/bf01141096, ISSN\u00a00025-5874, MR0260779\u00a0Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN\u00a0978-0-387-98428-5, MR1653294\u00a0Nagata, Masayoshi (1975), \u201cSome types of simple ring extensions\u201d, Houston J. Math. 1 (1):  131\u2013136, ISSN\u00a00362-1588, MR0382248, http:\/\/hjm.math.unizh.ch\/v001n1\/0131NAGATA.pdf\u00a0[\u30ea\u30f3\u30af\u5207\u308c]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki5\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki5\/archives\/6944#breadcrumbitem","name":"\u4e3b\u30a4\u30c7\u30a2\u30eb\u306b\u95a2\u3059\u308b\u6607\u9396\u6761\u4ef6 – Wikipedia"}}]}]