[{"data":1,"prerenderedAt":350},["ShallowReactive",2],{"$fJ4wOpQrJJKrkyvFJbZ54eXSi2XzuRCTHPhSI8SJLQY8":3,"$fDiwR3u-0wEQVuc_-YY7TLM24h7b7j_jhtuRrftg0Fng":47,"$fvBCk4GcXVpbEkci8IBsYSph7J64MZ6kLl2BvamoQgFg":62,"$fIuLrEhCuLGHK3A8wDAluG1OF8UqmlREfVPpUnaeFZrE":244,"requiredCourseNames_21026224":339,"opengraphListBase/3EuI8kFBpDDo5FsndhPlHUdRweMlwxfWGpWkF6g627A":340,"opengraphListBase/Vlf9y5q96adxz4kxFYK-dxUJyxXf5foSlcbo6d6SzRI":341,"opengraphListBase/FDONXej5kq_8ORVMgxYFC1LASZIJ8ddrf0_LEoYMacA":342,"opengraphListBase/qxEeKRjBAmZ5V4RvaKMMymltaWb1Li6oTtvvP5cNVG4":343,"opengraphListBase/U8KBVWrFPLCNw2ps7Zh5CN60sPeoNKEuVvzVdR-ndkQ":344,"opengraphListBase/ebdq_2ujsNJ3r8j3McYy0YN2CBkZXxl-F2IhDedJr6c":345,"opengraphListBase/XQJweEPW3VCv6wKKNZXFfM-QsB5yNmH3gVLiml3Y2g8":346,"opengraphListBase/5b4qQs5gAxOWt6IeSf3f0w6HEEhc2YM4GHjqptJXHHE":347,"opengraphListBase/-FSQ1fr65IiMv5LBCOA2-MFrACJo3x-FSiZz9-MtYE0":348,"opengraphListBase/KG94py1d0PizQZ9ZdUYNjcd4Dwr0IP4p_jj9phmi93A":349},{"timeTableCode":4,"year":5,"title":6,"englishTitle":7,"url":8,"semester":9,"method":10,"credits":11,"lecturers":12,"office":13,"email":14,"website":15,"lastUpdate":16,"theme":17,"prerequisites":18,"recommends":19,"materials":20,"outline":21,"practicalContent":10,"selfStudy":22,"evaluation":23,"officeHours":24,"message":25,"others":15,"mainCategory":26,"numberingCodes":31,"grades":36,"periods":37,"keywords":39,"mainDepartment":10,"creditCategories":42,"requiredCourses":43,"requiredCategories":44,"requiredJudgements":45,"departmentIds":46},"21026224",2026,"現代代数学(大学院連携科目)","Modern Algebra","https://kyoumu.office.uec.ac.jp/syllabus/2026/31/31_21026224.html","後学期","",2,"大野 真裕","東1-411","masahiro-ohno@uec.ac.jp","なし","2026-03-31T06:44:44.000Z","環上の加群の基礎について講義する. 特に, 単項イデアル整域(実際にはより強くユークリッド整域)上の単因子論について講義する. \n\nThis lecture provides basics of modules over a ring.","線形代数学第一, 同第二\n\nLinear algebra I and II","大学での代数学系の科目\n\nSome very elementary level lectures of algebras.","教科書:堀田良之著「代数入門」裳華房 第1章6節以降と第2章\n\nThe textbook is the book above written in Japanese.","第1回:環と環の間の準同型写像と関連概念(Rings and Homomorphisms of rings, and related concepts)\n(可換)環, 有理整数環, 多項式環, 環の間の準同型写像とその核と像, イデアル, 部分環\n第2回:単項イデアル整域と関連概念(Principal ideal domains, and related concepts)\n整域, イデアルの生成系, 単項イデアル整域, ユークリッド整域, ユークリッドの互除法\n第3回:剰余環と関連概念(Quotient rings, and related concepts)\n剰余環と環の準同型定理\n第4回:素イデアルと関連概念(Prime ideals, and related concepts)\n素イデアル, 極大イデアル\n第5回:中国式剰余定理と関連概念(Chinese Remainder Theorem, and related concepts)\n環の直積, 最大公約元, 2つのイデアルが互いに素, 中国式剰余定理\n第6回:環上の加群と関連概念(Modules over a ring, and related concepts)\n可換環上の加群, 加群の直積と直和, 自由加群\n第7回:環上の加群の間の準同型写像と関連概念(Homomorphisms of modules over a ring, and related concepts)\n環上の加群の間の準同型写像とその核と像, 部分加群\n第8回:剰余加群と関連概念(Quotient modules, and related concepts)\n剰余加群と加群の準同型定理\n第9回:生成系と関連概念(Generators and related concepts)\n生成系, 可換環上の自由加群の基底\n第10回:ネーター環上の有限生成加群と関連概念(Finitely generated modules over a noetherian ring, and related concepts)\nネーター環, 有限生成加群, ネーター加群\n第11回:ユークリッド整域上の行および列基本変形と関連概念(Elementary row or column operations over a euclidean domain, and related concepts)\nユークリッド整域上の行および列基本変形, 基本行列, 行列の単因子\n第12回:ねじれ部分加群と関連概念(Torsion submodules, and related concepts)\nねじれ部分加群, ユークリッド整域(単項イデアル整域)上の有限生成加群の構造定理\n第13回:ジョルダンの標準形と関連概念(Jordan canonical forms, and related concepts)\nジョルダンの標準形とその計算例\n第14回:可換環のスペクトラム(The spectrum of a commutative ring, and related concepts)\n可換環のスペクトラム, 有限生成加群の台\n第15回:まとめと復習(Summary)","授業時間外での学習は必須です. ","環および環上の加群の話の基礎によくある(形式的な)証明を身につけているか, 簡単な証明が書けるか等を基準に, 授業中に課したレポートの出来, および, 授業中の取り組み具合等で総合評価する. ","随時受け付ける. ","整数を係数とする一次方程式の整数解などは, ユークリッドの互除法などとともに, どこかの段階で学ばれた方も多いだろう. 一方, 実数を係数とする一次方程式の実数解などは, 線形代数学で学んだ. ユークリッド整域上の単因子論は, この2つの一般化にあたり, 高校で教える際にも, この講義の内容をきちんと身につけていれば, より良く問題を見通すことができるようになる. また, この講義の内容を身につければ, ジョルダンの標準形についても(実質的に)理解したことになる. 基本的に受講生の方の基礎知識等にあわせて講義するので, わからないことなどがあれば, 積極的に質問してください. ",{"id":27,"name":28,"supplement":29,"parentId":30},47,"自由科目",null,37,[32,33,34,35],"MTHb02a","MTHb02b","MTHb02e","MTHb02f",[],[38],"水5",[40,41],"Modules over a ring","環上の加群",[],[],[],[],[],[48,52,55,59],{"id":49,"name":50,"judgeAt":51},1,"2年次終了審査","2年次終了時",{"id":11,"name":53,"judgeAt":54},"卒業研究着手審査","3年次終了時",{"id":56,"name":57,"judgeAt":58},3,"卒業審査","4年次終了時",{"id":60,"name":61,"judgeAt":54},4,"輪講履修条件",[63,160,187,237,240],{"id":49,"name":64,"supplement":29,"children":65},"総合文化科目",[66,70,102,106,110,134,138,142,155],{"id":67,"name":68,"supplement":29,"children":69},6,"人文・社会科学科目",[],{"id":71,"name":72,"supplement":29,"children":73},7,"言語文化科目",[74,86,98],{"id":75,"name":76,"supplement":29,"children":77},15,"言語文化科目I",[78,82],{"id":79,"name":80,"supplement":29,"children":81},18,"言語文化基礎科目I",[],{"id":83,"name":84,"supplement":29,"children":85},19,"言語文化応用科目I",[],{"id":87,"name":88,"supplement":29,"children":89},16,"言語文化科目II",[90,94],{"id":91,"name":92,"supplement":29,"children":93},20,"言語文化基礎科目II",[],{"id":95,"name":96,"supplement":29,"children":97},21,"言語文化応用科目II",[],{"id":99,"name":100,"supplement":29,"children":101},17,"言語文化演習科目",[],{"id":103,"name":104,"supplement":29,"children":105},8,"健康・スポーツ科学科目",[],{"id":107,"name":108,"supplement":29,"children":109},9,"理工系教養科目",[],{"id":111,"name":112,"supplement":29,"children":113},10,"上級科目",[114,118,122,126,130],{"id":115,"name":116,"supplement":29,"children":117},22,"A類「文化と社会」",[],{"id":119,"name":120,"supplement":29,"children":121},23,"B類「言語によるコミュニケーション」",[],{"id":123,"name":124,"supplement":29,"children":125},24,"C類「異文化の理解」",[],{"id":127,"name":128,"supplement":29,"children":129},25,"D類「現代の科学」",[],{"id":131,"name":132,"supplement":29,"children":133},26,"E類「健康とスポーツの科学」",[],{"id":135,"name":136,"supplement":29,"children":137},11,"国際科目",[],{"id":139,"name":140,"supplement":29,"children":141},12,"特別講義",[],{"id":143,"name":144,"supplement":145,"children":146},13,"日本語・日本文化科目","外国人留学生のみ",[147,151],{"id":148,"name":149,"supplement":29,"children":150},27,"日本語",[],{"id":152,"name":153,"supplement":29,"children":154},28,"日本文化科目",[],{"id":156,"name":157,"supplement":158,"children":159},14,"健康科学科目","先端工学基礎課程のみ",[],{"id":11,"name":161,"supplement":29,"children":162},"実践教育科目",[163,167,171,175,179,183],{"id":164,"name":165,"supplement":29,"children":166},29,"初年次導入科目",[],{"id":168,"name":169,"supplement":29,"children":170},30,"データサイエンス科目",[],{"id":172,"name":173,"supplement":29,"children":174},31,"倫理・キャリア教育科目",[],{"id":176,"name":177,"supplement":29,"children":178},32,"技術英語科目",[],{"id":180,"name":181,"supplement":158,"children":182},33,"産学連携教育科目",[],{"id":184,"name":185,"supplement":158,"children":186},34,"技術者教養科目",[],{"id":56,"name":188,"supplement":29,"children":189},"専門科目",[190,194,213,230,234],{"id":191,"name":192,"supplement":29,"children":193},35,"理数基礎科目",[],{"id":195,"name":196,"supplement":29,"children":197},36,"類共通基礎科目",[198,202,206,210],{"id":199,"name":200,"supplement":29,"children":201},40,"必修科目",[],{"id":203,"name":204,"supplement":29,"children":205},41,"選択必修科目",[],{"id":207,"name":208,"supplement":29,"children":209},42,"選択科目",[],{"id":211,"name":28,"supplement":29,"children":212},43,[],{"id":30,"name":214,"supplement":29,"children":215},"類専門科目",[216,219,222,225,227],{"id":217,"name":200,"supplement":29,"children":218},44,[],{"id":220,"name":204,"supplement":29,"children":221},45,[],{"id":223,"name":208,"supplement":29,"children":224},46,[],{"id":27,"name":28,"supplement":29,"children":226},[],{"id":228,"name":136,"supplement":29,"children":229},48,[],{"id":231,"name":232,"supplement":158,"children":233},38,"専門基礎科目",[],{"id":235,"name":188,"supplement":158,"children":236},39,[],{"id":60,"name":238,"supplement":29,"children":239},"共通単位",[],{"id":241,"name":242,"supplement":29,"children":243},5,"教職科目",[],[245,331],{"id":246,"name":247,"hidden":248,"children":249},1000,"昼間コース",true,[250,275,307],{"id":251,"name":252,"hidden":253,"children":254},1100,"I類 (情報系)",false,[255,259,263,267,271],{"id":256,"name":257,"hidden":253,"children":258},1101,"メディア情報学プログラム",[],{"id":260,"name":261,"hidden":253,"children":262},1102,"経営・社会情報学プログラム",[],{"id":264,"name":265,"hidden":253,"children":266},1103,"情報数理工学プログラム",[],{"id":268,"name":269,"hidden":253,"children":270},1104,"コンピュータサイエンスプログラム",[],{"id":272,"name":273,"hidden":253,"children":274},1105,"デザイン思考・データサイエンスプログラム",[],{"id":276,"name":277,"hidden":253,"children":278},1200,"II類 (融合系)",[279,295],{"id":280,"name":281,"hidden":253,"children":282},1210,"情報工学エリア",[283,287,291],{"id":284,"name":285,"hidden":253,"children":286},1211,"セキュリティ情報学プログラム",[],{"id":288,"name":289,"hidden":253,"children":290},1212,"情報通信工学プログラム",[],{"id":292,"name":293,"hidden":253,"children":294},1213,"電子情報学プログラム",[],{"id":296,"name":297,"hidden":253,"children":298},1220,"メカトロニクスエリア",[299,303],{"id":300,"name":301,"hidden":253,"children":302},1221,"計測・制御システムプログラム",[],{"id":304,"name":305,"hidden":253,"children":306},1222,"先端ロボティクスプログラム",[],{"id":308,"name":309,"hidden":253,"children":310},1300,"III類 (理工系)",[311,315,319,323,327],{"id":312,"name":313,"hidden":253,"children":314},1301,"機械システムプログラム",[],{"id":316,"name":317,"hidden":253,"children":318},1302,"電子工学プログラム",[],{"id":320,"name":321,"hidden":253,"children":322},1303,"光工学プログラム",[],{"id":324,"name":325,"hidden":253,"children":326},1304,"物理工学プログラム",[],{"id":328,"name":329,"hidden":253,"children":330},1305,"化学生命工学プログラム",[],{"id":332,"name":333,"hidden":248,"children":334},2000,"夜間主コース",[335],{"id":336,"name":337,"hidden":253,"children":338},2001,"先端工学基礎課程",[],[],{},{},{},{},{},{},{},{},{},{},1775583432280]