[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki5\/archives\/7817#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki5\/archives\/7817","headline":"\u74b0\u306e\u6839\u57fa &#8211; Wikipedia","name":"\u74b0\u306e\u6839\u57fa &#8211; Wikipedia","description":"\u74b0\u8ad6\u3068\u3044\u3046\u6570\u5b66\u306e\u5206\u91ce\u306b\u304a\u3044\u3066\u3001\u74b0\u306e\u6839\u57fa (radical of a ring) \u306f\u74b0\u306e\u300c\u60aa\u3044\u300d\u5143\u304b\u3089\u306a\u308b\u30a4\u30c7\u30a2\u30eb\u3067\u3042\u308b\u3002 \u6839\u57fa\u306e\u6700\u521d\u306e\u4f8b\u306f\u51aa\u96f6\u6839\u57fa\u3067\u3042\u3063\u305f\u3002\u3053\u308c\u306f (Wedderburn 1908) \u306e\u30b5\u30b8\u30a7\u30b9\u30c1\u30e7\u30f3\u306b\u57fa\u3065\u3044\u3066\u3001(K\u00f6the 1930) \u3067\u5c0e\u5165\u3055\u308c\u305f\u3002\u6b21\u306e\u6570\u5e74\u9593\u3067\u3044\u304f\u3064\u304b\u306e\u4ed6\u306e\u6839\u57fa\u304c\u767a\u898b\u3055\u308c\u305f\u3002\u305d\u308c\u3089\u306e\u3046\u3061\u6700\u3082\u91cd\u8981\u306a\u4f8b\u306f\u30b8\u30e3\u30b3\u30d6\u30bd\u30f3\u6839\u57fa\u3067\u3042\u308b\u3002\u6839\u57fa\u306e\u4e00\u822c\u8ad6\u306f (Amitsur\u00a01952, 1954, 1954b) \u3068 Kurosh (1953) \u306b\u3088\u3063\u3066\u72ec\u7acb\u306b\u5b9a\u7fa9\u3055\u308c\u305f\u3002 \u6839\u57fa\u306e\u7406\u8ad6\u306b\u304a\u3044\u3066\u3001\u74b0\u306f\u901a\u5e38\u7d50\u5408\u7684\u306a\u3082\u306e\u3092\u8003\u3048\u308b\u304c\u3001\u53ef\u63db\u3067\u3042\u308b\u5fc5\u8981\u306f\u306a\u304f\u3001\u5358\u4f4d\u5143\u3092\u3082\u3064\u5fc5\u8981\u306f\u306a\u3044\u3002\u7279\u306b\u3001\u74b0\u306e\u3059\u3079\u3066\u306e\u30a4\u30c7\u30a2\u30eb\u306f\u307e\u305f\u74b0\u3067\u3042\u308b\u3002","datePublished":"2017-12-31","dateModified":"2017-12-31","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\/526fa7aee07d85f6c8be4ede2d45b99f42ee1b35","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/526fa7aee07d85f6c8be4ede2d45b99f42ee1b35","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/jp\/wiki5\/archives\/7817","about":["Wiki"],"wordCount":2924,"articleBody":"\u74b0\u8ad6\u3068\u3044\u3046\u6570\u5b66\u306e\u5206\u91ce\u306b\u304a\u3044\u3066\u3001\u74b0\u306e\u6839\u57fa (radical of a ring) \u306f\u74b0\u306e\u300c\u60aa\u3044\u300d\u5143\u304b\u3089\u306a\u308b\u30a4\u30c7\u30a2\u30eb\u3067\u3042\u308b\u3002\u6839\u57fa\u306e\u6700\u521d\u306e\u4f8b\u306f\u51aa\u96f6\u6839\u57fa\u3067\u3042\u3063\u305f\u3002\u3053\u308c\u306f (Wedderburn 1908) \u306e\u30b5\u30b8\u30a7\u30b9\u30c1\u30e7\u30f3\u306b\u57fa\u3065\u3044\u3066\u3001(K\u00f6the 1930) \u3067\u5c0e\u5165\u3055\u308c\u305f\u3002\u6b21\u306e\u6570\u5e74\u9593\u3067\u3044\u304f\u3064\u304b\u306e\u4ed6\u306e\u6839\u57fa\u304c\u767a\u898b\u3055\u308c\u305f\u3002\u305d\u308c\u3089\u306e\u3046\u3061\u6700\u3082\u91cd\u8981\u306a\u4f8b\u306f\u30b8\u30e3\u30b3\u30d6\u30bd\u30f3\u6839\u57fa\u3067\u3042\u308b\u3002\u6839\u57fa\u306e\u4e00\u822c\u8ad6\u306f (Amitsur\u00a01952, 1954, 1954b) \u3068 Kurosh (1953) \u306b\u3088\u3063\u3066\u72ec\u7acb\u306b\u5b9a\u7fa9\u3055\u308c\u305f\u3002\u6839\u57fa\u306e\u7406\u8ad6\u306b\u304a\u3044\u3066\u3001\u74b0\u306f\u901a\u5e38\u7d50\u5408\u7684\u306a\u3082\u306e\u3092\u8003\u3048\u308b\u304c\u3001\u53ef\u63db\u3067\u3042\u308b\u5fc5\u8981\u306f\u306a\u304f\u3001\u5358\u4f4d\u5143\u3092\u3082\u3064\u5fc5\u8981\u306f\u306a\u3044\u3002\u7279\u306b\u3001\u74b0\u306e\u3059\u3079\u3066\u306e\u30a4\u30c7\u30a2\u30eb\u306f\u307e\u305f\u74b0\u3067\u3042\u308b\u3002\u6839\u57fa\u30af\u30e9\u30b9 (radical class)\uff08\u6839\u57fa\u6027\u8cea (radical property) \u3084\u5358\u306b\u6839\u57fa (radical) \u3068\u3082\u547c\u3070\u308c\u308b\uff09\u306f\u5358\u4f4d\u5143\u306e\u5b58\u5728\u3092\u4eee\u5b9a\u3057\u306a\u3044\u74b0\u306e\u30af\u30e9\u30b9 \u03c3 \u3067\u3042\u3063\u3066\u3001\u4ee5\u4e0b\u3092\u6e80\u305f\u3059\u3082\u306e\u3067\u3042\u308b\uff1a(1) \u03c3 \u306b\u5165\u3063\u3066\u3044\u308b\u74b0\u306e\u6e96\u540c\u578b\u50cf\u306f\u307e\u305f \u03c3 \u306b\u5165\u308b\u3002(2) \u3059\u3079\u3066\u306e\u74b0 R \u306f \u03c3 \u306b\u5165\u3063\u3066\u3044\u308b\u3059\u3079\u3066\u306e\u4ed6\u306e\u30a4\u30c7\u30a2\u30eb\u3092\u542b\u3080 \u03c3 \u306b\u5165\u3063\u3066\u3044\u308b\u30a4\u30c7\u30a2\u30eb S(R) \u3092\u542b\u3080\u3002(3) S(R\/S(R))\u00a0=\u00a00\u3002\u30a4\u30c7\u30a2\u30eb S(R) \u306f R \u306e\u6839\u57fa\u3001\u3042\u308b\u3044\u306f \u03c3-\u6839\u57fa\u3068\u547c\u3070\u308c\u308b\u3002\u305d\u306e\u3088\u3046\u306a\u6839\u57fa\u306e\u7814\u7a76\u306f torsion theory \u3068\u547c\u3070\u308c\u308b\u3002\u74b0\u306e\u4efb\u610f\u306e\u30af\u30e9\u30b9 \u03b4 \u306b\u5bfe\u3057\u3066\u3001\u305d\u308c\u3092\u542b\u3080\u6700\u5c0f\u306e\u6839\u57fa\u30af\u30e9\u30b9 L\u03b4 \u304c\u5b58\u5728\u3057\u3001\u03b4 \u306e lower radical \u3068\u547c\u3076\u3002\u4f5c\u7528\u7d20 L \u3092 lower radical operator \u3068\u8a00\u3046\u3002\u74b0\u306e\u30af\u30e9\u30b9\u306f\u30af\u30e9\u30b9\u306b\u5165\u3063\u3066\u3044\u308b\u74b0\u306e\u3059\u3079\u3066\u306e 0 \u3067\u306a\u3044\u30a4\u30c7\u30a2\u30eb\u304c\u30af\u30e9\u30b9\u306b\u5165\u308b 0 \u3067\u306a\u3044\u50cf\u3092\u3082\u3064\u3068\u304d\u6b63\u5247 (regular) \u3068\u547c\u3070\u308c\u308b\u3002\u74b0\u306e\u3059\u3079\u3066\u306e\u6b63\u5247\u30af\u30e9\u30b9 \u03b4 \u306b\u5bfe\u3057\u3066\u3001\u6700\u5927\u306e\u6839\u57fa\u30af\u30e9\u30b9 U\u03b4 \u304c\u5b58\u5728\u3057\u3001\u03b4 \u306e upper radical \u3068\u547c\u3070\u308c\u3001\u03b4 \u3068\u306e\u5171\u901a\u90e8\u5206\u306f 0 \u3067\u3042\u308b\u3002\u4f5c\u7528\u7d20 U \u306f upper radical operator \u3068\u547c\u3070\u308c\u308b\u3002\u74b0\u306e\u30af\u30e9\u30b9\u306f\u3001\u30af\u30e9\u30b9\u306b\u5165\u3063\u3066\u3044\u308b\u74b0\u306e\u3059\u3079\u3066\u306e\u30a4\u30c7\u30a2\u30eb\u304c\u307e\u305f\u30af\u30e9\u30b9\u306b\u5c5e\u3057\u3066\u3044\u308b\u3068\u304d\u306b\u3001\u907a\u4f1d\u7684 (hereditary) \u3068\u547c\u3070\u308c\u308b\u3002Table of Contents\u30b8\u30e3\u30b3\u30d6\u30bd\u30f3\u6839\u57fa[\u7de8\u96c6]Baer\u6839\u57fa[\u7de8\u96c6]upper nil radical \u3042\u308b\u3044\u306f K\u00f6the radical[\u7de8\u96c6]\u7279\u7570\u6839\u57fa[\u7de8\u96c6]\u30ec\u30f4\u30a3\u30c4\u30ad\u6839\u57fa[\u7de8\u96c6]\u30d6\u30e9\u30a6\u30f3\u2013\u30de\u30c3\u30b3\u30a4\u6839\u57fa[\u7de8\u96c6]\u30d5\u30a9\u30f3\u30fb\u30ce\u30a4\u30de\u30f3\u6b63\u5247\u6839\u57fa[\u7de8\u96c6]\u30a2\u30eb\u30c6\u30a3\u30f3\u6839\u57fa[\u7de8\u96c6]\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u30b8\u30e3\u30b3\u30d6\u30bd\u30f3\u6839\u57fa[\u7de8\u96c6]R \u3092\u53ef\u63db\u3068\u306f\u9650\u3089\u306a\u3044\u4efb\u610f\u306e\u74b0\u3068\u3059\u308b\u3002R \u306e\u30b8\u30e3\u30b3\u30d6\u30bd\u30f3\u6839\u57fa (Jacobson radical of R) \u306f\u3059\u3079\u3066\u306e\u5358\u7d14\u53f3 R-\u52a0\u7fa4\u306e\u96f6\u5316\u30a4\u30c7\u30a2\u30eb\u306e\u5171\u901a\u90e8\u5206\u3067\u3042\u308b\u3002  \u30b8\u30e3\u30b3\u30d6\u30bd\u30f3\u6839\u57fa\u306e\u3044\u304f\u3064\u304b\u306e\u540c\u5024\u306a\u7279\u5fb4\u3065\u3051\u304c\u5b58\u5728\u3059\u308b\u3002\u4f8b\u3048\u3070\uff1aJ(R) \u306f R \u306e\u6b63\u5247\u6975\u5927\u53f3\uff08\u3042\u308b\u3044\u306f\u5de6\uff09\u30a4\u30c7\u30a2\u30eb\u306e\u5171\u901a\u90e8\u5206\u3067\u3042\u308b\u3002J(R) \u306f R \u306e\u3059\u3079\u3066\u306e\u53f3\uff08\u3042\u308b\u3044\u306f\u5de6\uff09\u539f\u59cb\u30a4\u30c7\u30a2\u30eb\u306e\u5171\u901a\u90e8\u5206\u3067\u3042\u308b\u3002J(R) \u306f R \u306e\u6975\u5927\u53f3\uff08\u3042\u308b\u3044\u306f\u5de6\uff09\u6e96\u6b63\u5247\u53f3\uff08resp. \u5de6\uff09\u30a4\u30c7\u30a2\u30eb\u3067\u3042\u308b\u3002\u51aa\u96f6\u6839\u57fa\u306e\u3088\u3046\u306b\u3001\u3053\u306e\u5b9a\u7fa9\u3092\u4efb\u610f\u306e\u4e21\u5074\u30a4\u30c7\u30a2\u30eb I \u306b\u62e1\u5f35\u3059\u308b\u3053\u3068\u304c J(I) \u3092\u5c04\u5f71 R\u2192R\/I \u306e\u4e0b\u3067\u306e J(R\/I) \u306e\u539f\u50cf\u3068\u5b9a\u7fa9\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066\u3067\u304d\u308b\u3002R \u304c\u53ef\u63db\u3067\u3042\u308c\u3070\u3001\u30b8\u30e3\u30b3\u30d6\u30bd\u30f3\u6839\u57fa\u306f\u5e38\u306b\u51aa\u96f6\u6839\u57fa\u3092\u542b\u3080\u3002\u74b0 R \u304c\u6709\u9650\u751f\u6210 Z-\u4ee3\u6570\u3067\u3042\u308c\u3070\u3001\u51aa\u96f6\u6839\u57fa\u306f\u30b8\u30e3\u30b3\u30d6\u30bd\u30f3\u6839\u57fa\u306b\u7b49\u3057\u304f\u3001\u3088\u308a\u4e00\u822c\u7684\u306b\uff1a \u4efb\u610f\u306e\u30a4\u30c7\u30a2\u30eb I \u306e\u6839\u57fa\u306f I \u3092\u542b\u3080 R \u306e\u3059\u3079\u3066\u306e\u6975\u5927\u30a4\u30c7\u30a2\u30eb\u306e\u5171\u901a\u90e8\u5206\u306b\u5e38\u306b\u7b49\u3057\u3044\u3002\u3053\u308c\u306f R \u304c\u30b8\u30e3\u30b3\u30d6\u30bd\u30f3\u74b0\u3067\u3042\u308b\u3068\u8a00\u3063\u3066\u3044\u308b\u3002Baer\u6839\u57fa[\u7de8\u96c6]\u74b0 R \u306e Baer \u6839\u57fa\u306fR \u306e\u7d20\u30a4\u30c7\u30a2\u30eb\u5168\u90e8\u306e\u5171\u901a\u90e8\u5206\u3067\u3042\u308b\u3002\u540c\u5024\u3060\u304c\u3001\u305d\u308c\u306fR \u306e\u6700\u5c0f\u306e\u534a\u7d20\u30a4\u30c7\u30a2\u30eb\u3067\u3042\u308b\u3002Baer \u6839\u57fa\u306f\u51aa\u96f6\u74b0\u306e\u30af\u30e9\u30b9\u306e lower radical \u3067\u3042\u308b\u3002\u6b21\u306e\u3088\u3046\u306b\u3082\u547c\u3070\u308c\u308b\u3002&#8221;lower nilradical&#8221;\uff08\u305d\u3057\u3066 Nil\u2217R \u3068\u8868\u8a18\u3055\u308c\u308b\uff09\u3001&#8221;prime radical&#8221;\u3001&#8221;Baer-McCoy radical&#8221;\u3002Baer \u6839\u57fa\u306e\u3059\u3079\u3066\u306e\u5143\u306f\u51aa\u96f6\u3067\u3042\u308a\u3001\u305d\u306e\u305f\u3081\u305d\u308c\u306f\u51aa\u96f6\u5143\u30a4\u30c7\u30a2\u30eb\uff08\u82f1\u8a9e\u7248\uff09\u3067\u3042\u308b\u3002  \u53ef\u63db\u74b0\u306b\u5bfe\u3057\u3066\u3001\u3053\u308c\u306f\u5358\u306b\u51aa\u96f6\u6839\u57fa\u3067\u3042\u308a\u3001\u30a4\u30c7\u30a2\u30eb\u306e\u6839\u57fa\u306e\u5b9a\u7fa9\u304c\u5bc6\u63a5\u306b\u5f93\u3046\u3002upper nil radical \u3042\u308b\u3044\u306f K\u00f6the radical[\u7de8\u96c6]\u74b0 R \u306e\u51aa\u96f6\u5143\u30a4\u30c7\u30a2\u30eb\uff08\u82f1\u8a9e\u7248\uff09\u5168\u4f53\u306e\u548c\u306f upper nilradical Nil*R \u3042\u308b\u3044\u306f K\u00f6the radical \u3067\u3042\u308a\u3001R \u306e\u552f\u4e00\u306e\u6700\u5927\u306e\u51aa\u96f6\u5143\u30a4\u30c7\u30a2\u30eb\u3067\u3042\u308b\u3002K\u00f6the\u306e\u4e88\u60f3\u306f\u4efb\u610f\u306e\u5de6\u51aa\u96f6\u5143\u30a4\u30c7\u30a2\u30eb\u304c\u305d\u306e nilradical \u306b\u5165\u308b\u304b\u3069\u3046\u304b\u3092\u554f\u3046\u3002\u7279\u7570\u6839\u57fa[\u7de8\u96c6]\uff08\u975e\u53ef\u63db\u3067\u3082\u3088\u3044\uff09\u74b0\u306e\u5143\u306f\u3042\u308b\u672c\u8cea\u5de6\u30a4\u30c7\u30a2\u30eb\u3092\u96f6\u5316\u3059\u308b\u3068\u304d\u306b\u5de6\u7279\u7570 (singular) \u3067\u3042\u308b\u3068\u8a00\u3046\u3002\u3064\u307e\u308a\u3001r \u304c\u5de6\u7279\u7570\u3068\u306f\u3001\u3042\u308b\u672c\u8cea\u5de6\u30a4\u30c7\u30a2\u30eb I \u306b\u5bfe\u3057\u3066 Ir = 0 \u3068\u3044\u3046\u3053\u3068\u3067\u3042\u308b\u3002\u74b0 R \u306e\u5de6\u7279\u7570\u5143\u5168\u4f53\u306e\u96c6\u5408\u306f\u4e21\u5074\u30a4\u30c7\u30a2\u30eb\u3067\u3042\u308a\u3001\u5de6\u7279\u7570\u30a4\u30c7\u30a2\u30eb\u3068\u547c\u3070\u308c\u3001Z(RR){displaystyle {mathcal {Z}}(_{R}R),} \u3068\u8868\u8a18\u3055\u308c\u308b\u3002  N\/Z(RR)=Z(R\/Z(RR)R\/Z(RR)){displaystyle N\/{mathcal {Z}}(_{R}R)={mathcal {Z}}(_{R\/{mathcal {Z}}(_{R}R)}R\/{mathcal {Z}}(_{R}R)),} \u3067\u3042\u308b\u3088\u3046\u306a R \u306e\u30a4\u30c7\u30a2\u30eb N \u306f Z2(RR){displaystyle {mathcal {Z}}_{2}(_{R}R)} \u3068\u8868\u8a18\u3055\u308c\u3001R \u306e\u7279\u7570\u6839\u57fa (singular radical) \u3042\u308b\u3044\u306f\u30b4\u30eb\u30c7\u30a3\u30fc\u30c8\u30fc\u30b7\u30e7\u30f3 (Goldie torsion) \u3068\u547c\u3070\u308c\u308b\u3002\u7279\u7570\u6839\u57fa\u306f\u7d20\u6839\u57fa\uff08\u53ef\u63db\u74b0\u306e\u5834\u5408\u306b\u306f\u3053\u308c\u306f\u51aa\u96f6\u6839\u57fa\u3067\u3042\u308b\uff09\u3092\u542b\u3080\u304c\u3001\u53ef\u63db\u74b0\u306e\u5834\u5408\u3067\u3059\u3089\u3001\u305d\u308c\u3092\u771f\u306b\u542b\u3080\u304b\u3082\u3057\u308c\u306a\u3044\u3002\u3057\u304b\u3057\u306a\u304c\u3089\u3001\u30cd\u30fc\u30bf\u30fc\u74b0\u306e\u7279\u7570\u6839\u57fa\u306f\u5e38\u306b\u51aa\u96f6\u3067\u3042\u308b\u3002\u30ec\u30f4\u30a3\u30c4\u30ad\u6839\u57fa[\u7de8\u96c6]Levitzki \u6839\u57fa\u306f\u6700\u5927\u306e\u5c40\u6240\u7684\u51aa\u96f6\u30a4\u30c7\u30a2\u30eb\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u3001\u7fa4\u8ad6\u306eHirsch\u2013Plotkin \u6839\u57fa\uff08\u82f1\u8a9e\u7248\uff09\u3068\u30a2\u30ca\u30ed\u30ac\u30b9\u3067\u3042\u308b\u3002\u74b0\u304c\u30cd\u30fc\u30bf\u30fc\u3067\u3042\u308c\u3070\u3001Levitzki \u6839\u57fa\u306f\u305d\u308c\u81ea\u8eab\u51aa\u96f6\u30a4\u30c7\u30a2\u30eb\u3067\u3042\u308a\u3001\u305d\u308c\u3086\u3048\u552f\u4e00\u306e\u6700\u5927\u5de6\u3001\u53f3\u3001\u3042\u308b\u3044\u306f\u4e21\u5074\u51aa\u96f6\u30a4\u30c7\u30a2\u30eb\u3067\u3042\u308b\u3002\u30d6\u30e9\u30a6\u30f3\u2013\u30de\u30c3\u30b3\u30a4\u6839\u57fa[\u7de8\u96c6]Brown\u2013McCoy \u6839\u57fa\uff08\u30d0\u30ca\u30c3\u30cf\u4ee3\u6570\u306e\u7406\u8ad6\u3067\u306f\u5f37\u6839\u57fa (strong radical) \u3068\u547c\u3070\u308c\u308b\uff09\u306f\u4ee5\u4e0b\u306e\u65b9\u6cd5\u306e\u4efb\u610f\u3067\u5b9a\u7fa9\u3067\u304d\u308b\uff1a\u6975\u5927\u4e21\u5074\u30a4\u30c7\u30a2\u30eb\u5168\u4f53\u306e\u5171\u901a\u90e8\u5206\u3059\u3079\u3066\u306e\u6975\u5927\u30e2\u30b8\u30e5\u30e9\u30fc\u30a4\u30c7\u30a2\u30eb\u306e\u5171\u901a\u90e8\u5206\u5358\u4f4d\u5143\u3092\u3082\u3064\u3059\u3079\u3066\u306e\u5358\u7d14\u74b0\u306e\u30af\u30e9\u30b9\u306e upper radicalBrown\u2013McCoy \u6839\u57fa\u306f 1 \u3092\u3082\u3064\u7d50\u5408\u7684\u74b0\u3088\u308a\u3082\u306f\u308b\u304b\u306b\u4e00\u822c\u7684\u306a\u8a2d\u5b9a\u3067\u7814\u7a76\u3055\u308c\u308b\u3002\u30d5\u30a9\u30f3\u30fb\u30ce\u30a4\u30de\u30f3\u6b63\u5247\u6839\u57fa[\u7de8\u96c6]\u30d5\u30a9\u30f3\u30fb\u30ce\u30a4\u30de\u30f3\u6b63\u5247\u74b0\u306f\u74b0 A\uff08\u5358\u4f4d\u5143\u3092\u6301\u305f\u306a\u3044\u975e\u53ef\u63db\u74b0\u3067\u3088\u3044\uff09\u3067\u3042\u3063\u3066\u3001\u3059\u3079\u3066\u306e a \u306b\u5bfe\u3057\u3066\u3042\u308b b \u304c\u5b58\u5728\u3057\u3066  a = aba \u3068\u306a\u308b\u3088\u3046\u306a\u3082\u306e\u3067\u3042\u308b\u3002\u30d5\u30a9\u30f3\u30fb\u30ce\u30a4\u30de\u30f3\u6b63\u5247\u74b0\u306f\u6839\u57fa\u30af\u30e9\u30b9\u3092\u306a\u3059\u3002\u305d\u308c\u306f\u53ef\u9664\u4ee3\u6570\u4e0a\u306e\u3059\u3079\u3066\u306e\u884c\u5217\u74b0\u3092\u542b\u3080\u304c\u3001\u51aa\u96f6\u5143\u74b0 (nil ring) \u306f\u5168\u304f\u542b\u307e\u306a\u3044\u3002\u30a2\u30eb\u30c6\u30a3\u30f3\u6839\u57fa[\u7de8\u96c6]\u30a2\u30eb\u30c6\u30a3\u30f3\u6839\u57fa (Artinian radical) \u306f\u901a\u5e38\u4e21\u5074\u30cd\u30fc\u30bf\u30fc\u74b0\u306b\u5bfe\u3057\u3066\u30a2\u30eb\u30c6\u30a3\u30f3\u52a0\u7fa4\u3067\u3042\u308b\u3059\u3079\u3066\u306e\u53f3\u30a4\u30c7\u30a2\u30eb\u306e\u548c\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u5b9a\u7fa9\u306f\u5de6\u53f3\u5bfe\u79f0\u7684\u3067\u3042\u308a\u3001\u5b9f\u969b\u74b0\u306e\u4e21\u5074\u30a4\u30c7\u30a2\u30eb\u3092\u751f\u307f\u51fa\u3059\u3002\u3053\u306e\u6839\u57fa\u306f (Chatters 1980) \u3067\u6982\u8aac\u3055\u308c\u3066\u3044\u308b\u3088\u3046\u306b\u3001\u30cd\u30fc\u30bf\u30fc\u74b0\u306e\u7814\u7a76\u306b\u304a\u3044\u3066\u91cd\u8981\u3067\u3042\u308b\u3002\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u74b0\u306e\u6839\u57fa\u3067\u306f\u306a\u3044\u6839\u57fa\u306e\u95a2\u9023\u3057\u305f\u4f7f\u7528\uff1a\u53c2\u8003\u6587\u732e[\u7de8\u96c6]Andrunakievich, V.A. (2001), &#8220;Radical of ring and algebras&#8221;,  in Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics, Springer, ISBN\u00a0978-1-55608-010-4\u3002Chatters, A. W.; Hajarnavis, C. R. (1980), Rings with chain conditions, Research Notes in Mathematics, 44, Boston, Mass.: Pitman (Advanced Publishing Program), pp.\u00a0vii+197, ISBN\u00a00-273-08446-1, MR590045\u00a0Divinsky, N. J. (1965), Rings and radicals, Mathematical Expositions No. 14, Toronto, Ont.: University of Toronto Press, MR0197489\u00a0Gardner, B. J.; Wiegandt, R. (2004), Radical theory of rings, Monographs and Textbooks in Pure and Applied Mathematics, 261, New York: Marcel Dekker Inc., ISBN\u00a0978-0-8247-5033-6, MR2015465\u00a0Goodearl, K. R. (1976), Ring theory, Marcel Dekker, ISBN\u00a0978-0-8247-6354-1, MR0429962\u00a0Gray, Mary (1970), A radical approach to algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., MR0265396\u00a0K\u00f6the, Gottfried (1930), \u201cDie Struktur der Ringe, deren Restklassenring nach dem Radikal vollst\u00e4ndig reduzibel ist\u201d, Mathematische Zeitschrift 32 (1):  161\u2013186, doi:10.1007\/BF01194626\u00a0Stenstr\u00f6m, Bo (1971), Rings and modules of quotients, Lecture Notes in Mathematics, 237, Berlin, New York: Springer-Verlag, doi:10.1007\/BFb0059904, ISBN\u00a0978-3-540-05690-4, MR0325663, Zbl\u00a00229.16003\u00a0Wiegandt, Richard (1974), Radical and semisimple classes of rings, Kingston, Ont.: Queen&#8217;s University, MR0349734\u00a0"},{"@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\/7817#breadcrumbitem","name":"\u74b0\u306e\u6839\u57fa &#8211; Wikipedia"}}]}]