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系統識別號 U0026-1701201512171300
論文名稱(中文) 元素個數小於等於8的環的分類
論文名稱(英文) Classification of finite rings whose order not exceed 8
校院名稱 成功大學
系所名稱(中) 數學系應用數學碩博士班
系所名稱(英) Department of Mathematics
學年度 103
學期 1
出版年 104
研究生(中文) 張世杰
研究生(英文) Shih-Jie, Chang
電子信箱 bfhaha@gmail.com
學號 L16004062
學位類別 碩士
語文別 英文
論文頁數 69頁
口試委員 指導教授-柯文峰
口試委員-黃柏嶧
口試委員-黃世昌
中文關鍵字 有限環  有限環的分類 
英文關鍵字 finite ring  classification of finite rings  cyclic ring 
學科別分類
中文摘要 在本篇論文中,我們將對元素個數為質數的環還有元素個數不超過8的環做分類。第一小節對元素個數為質數的環做分類。第二小節對元素個數為6的環做分類。第三小節對元素個數為4的環做分類。第四小節對元素個數為8的環做分類。
英文摘要 In this thesis, we classify the rings of prime order and the rings whose order does not exceed 8.
論文目次 0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1 Rings of Prime Order . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Rings of order 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Rings of order 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
..3.1 (R,+) = (Z_4,+) . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
..3.2 (R,+) = (Z_2+Z_2,+) . . . . . . . . . . . . . . . . . . . . . . . . . 16
....3.2.1 R has a two-sided annihilator a . . . . . . . . . . . . . . . . . 16
....3.2.2 R has no two-sided annihilator, but has a left annihilator . . . 16
....3.2.3 R has no two-sided annihilator and has no left annihilator . . . 17
......3.2.3.1 The set of all nilpotent elements is trivial . . . . . . . . 18
......3.2.3.2 The set of all nilpotent elements is nontrivial . . . . . . . 19

4 Rings of order 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
..4.1 (R,+) = (Z_8,+) . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
..4.2 (R,+) = (Z_2+Z_4,+) . . . . . . . . . . . . . . . . . . . . . . . . . 22
....4.2.1 R has a multiplicative identity . . . . . . . . . . . . . . . . . 22
....4.2.2 R has no multiplicative identity . . . . . . . . . . . . . . . . 30
......4.2.2.1 R is commutative . . . . . . . . . . . . . . . . . . . . . . 33
........4.2.2.1.1 R^3 neq 0. . . . . . . . . . . . . . . . . . . . . . . . 35
........4.2.2.1.2 R^3 = 0. . . . . . . . . . . . . . . . . . . . . . . . . 38
......4.2.2.2 R is noncommutative . . . . . . . . . . . . . . . . . . . . 41
........4.2.2.2.1 R has no nonzero idempotent . . . . . . . . . . . . . . 41
........4.2.2.2.2 R has a nonzero idempotent e . . . . . . . . . . . . . . 44
..4.3 (R,+) = (Z_2+Z_2+Z_2;+) . . . . . . . . . . . . . . . . . . . . . . 48
....4.3.1 |J(R)| = 1 . . . . . . . . . . . . . . . . . . . . . . . . . 49
....4.3.2 |J(R)| = 2 . . . . . . . . . . . . . . . . . . . . . . . . . 49
....4.3.3 |J(R)| = 4 . . . . . . . . . . . . . . . . . . . . . . . . . 55
......4.3.3.1 R has a multiplicative identity . . . . . . . . . . . . . 55
......4.3.3.2 R has no multiplicative identity . . . . . . . . . . . . 56
....4.3.4 |J(R)| = 8 . . . . . . . . . . . . . . . . . . . . . . . . . 59
......4.3.4.1 R^3 neq 0 . . . . . . . . . . . . . . . . . . . . . . . 59
......4.3.4.2 R^3 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . 59
........4.3.4.2.1 For all r in R, r^2 = 0 . . . . . . . . . . . . . . 60
........4.3.4.2.2 There exists r^2 in R, r^2 = 0 . . . . . . . . . . . 60

5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1 Structure Theorem of Finite Ring . . . . . . . . . . . . . . . . 61
5.2 The hypothesis of each lemma . . . . . . . . . . . . . . . . . . 61
5.3 Interdependence of some theorems . . . . . . . . . . . . . . . . 63
5.4 Comparison table of the GAP command between the ring
number in this thesis . . . . . . . . . . . . . . . . . . . . . . . . 64

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
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