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研究生： 張世杰
Chang, Shih-Jie,
論文名稱： 元素個數小於等於8的環的分類
Classification of finite rings whose order not exceed 8
指導教授： 柯文峰
Ke, Wen-Fong
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系應用數學碩博士班
Department of Mathematics
論文出版年： 2015
畢業學年度： 103
語文別： 英文
論文頁數： 69
中文關鍵詞： 有限環 、有限環的分類
外文關鍵詞： finite ring, classification of finite rings, cyclic ring
相關次數： 點閱：76  下載：12 
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    • 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。

    In this thesis, we classify the rings of prime order and the rings whose order does not exceed 8.

    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

    [1] Decompose your ring using idempotents. http://www.tricki.org/
    article/Decompose_your_ring_using_idempotents.

    [2] Paolo Alu . Algebra: Chapter 0. AMC, 10:12, 2009.

    [3] VG Antipkin and VP Elizarov. Rings of order p 3. Siberian
    Mathematical Journal, 23(4):457{464, 1982. The original article is
    available at http://www.rusdml.de/rdms/rimg/?PPN=PPN394039319_
    0023&DMDID=DMDLOG_0080.

    [4] Michael Axtell, Joe Stickles, and Wallace Trampbachls. Zero-divisor
    ideals and realizable zero-divisor graphs. Involve, a Journal of Math-
    ematics, 2(1):17{27, 2009. Available at http://www.math.uiuc.edu/
    ~pecheni2/pdfs/involve.pdf.

    [5] Phani Bhushan Bhattacharya, Surender Kumar Jain, and SR Nagpaul.
    Basic abstract algebra. Cambridge University Press, 1994.

    [6] Gilberto Bini and Flaminio Flamini. Finite commutative rings and their
    applications. Springer, 2002.

    [7] Gilberto Bini and Flamino Flamini. Finite commutative rings and
    their applications. http://www.researchgate.net/publication/
    235426680_Finite_Commutative_Rings_and_Their_Applications/
    file/d912f513b47087384a.pdf.

    [8] DM Bloom. Rings of order four. Amer. Math. Monthly, 71:918{920,
    1964. http://www.jstor.org/stable/2312421.

    [9] Warren K. Buck. Cyclic rings. Master's thesis, Eastern Illinois Uni-
    versity, 2004. Available at https://www.uni.illinois.edu/~wbuck/
    thesis.pdf.

    [10] David S Dummit and Richard M Foote. Abstract algebra. 2004.

    [11] Colin R Fletcher. Rings of small order. The Mathematical Gazette, pages
    9{22, 1980.

    [12] Alexandra Forsythe and Neal H McCoy. On the commutativ-
    ity of certain rings. Bulletin of the American Mathematical
    Society, 52(6):523{526, 1946. Available at http://www.ams.
    org/journals/bull/1946-52-06/S0002-9904-1946-08603-6/
    S0002-9904-1946-08603-6.pdf.

    [13] Joseph Gallian. Contemporary abstract algebra. Cengage Learning, 2009.

    [14] William J Gilbert and W Keith Nicholson. Modern algebra with
    applications, volume 66. John Wiley & Sons, 2004. Available at
    http://cs.ioc.ee/~margo/aat/Gilbert%20W.J.,%20Nicholson%
    20W.K.%20Modern%20algebra%20with%20applications%20(2ed.,%
    20Wiley,%202004)(ISBN%200471414514)(347s).pdf.

    [15] Israel Nathan Herstein. Noncommutative rings. Number 15. Cambridge
    University Press, 2005.

    [16] Jonathan K Hodge, Steven Schlicker, and Ted Sundstrom. Abstract
    Algebra: An Inquiry Based Approach. CRC Press, 2013.

    [17] Thomas W Hungerford. Algebra, volume 73 of graduate texts in math-
    ematics, 1980.

    [18] N Jacobson. Basic algebra, 2 , 2009.

    [19] Nathan Jacobson. Structure theory for algebraic algebras of bounded
    degree. Annals of Mathematics, pages 695{707, 1945. http://www.
    jstor.org/stable/1969205.

    [20] Nathan Jacobson. Basic algebra I. Courier Dover Publications, 2012.

    [21] Nikita A Karpenko. Hyperbolicity of hermitian forms over biquater-
    nion algebras. Linear Algebraic Groups and Related Structures (preprint
    server), 2009. Available at http://www.math.uiuc.edu/K-theory/
    0922/hyperrho.pdf.

    [22] Robert Leroy Kruse and David T Price. Nilpotent rings. Gordon and
    Breach New York, 1969.

    [23] Piotr A Krylov and Askar A Tuganbaev. Modules over discrete valuation
    domains, volume 43. Walter de Gruyter, 2008.

    [24] E. L. Lady. Lifting idempotents modulo a nil ideal. http://www.math.
    hawaii.edu/~lee/algebra/idempotent.pdf.

    [25] Tsi-Yuen Lam. A rst course in noncommutative rings. Springer Science
    & Business, 2013.

    [26] Tsit-Yuen Lam and Tsit-Yuen Lam. Exercises in classical ring theory,
    volume 2. Springer, 2003.

    [27] Ju-Si Lee. The restricted solution of ax + by = gcd (a; b). Tai-
    wanese Journal of Mathematics, 12(5):1191{1199, 2008. Available at
    http://journal.taiwanmathsoc.org.tw/index.php/TJM/article/
    view/642/512.

    [28] Neal Henry McCoy. The theory of rings. Macmillan New York, 1964.

    [29] Bernard R McDonald. Finite rings with identity. M. Dekker New York,
    1974.

    [30] W Keith Nicholson. Introduction to abstract algebra. John Wiley &
    Sons, 2012.

    [31] Mihnea Popa. Let r be a ring with unity (identity). show that every
    element of r is either a unit or a zero divisor if r is nite. http://
    homepages.math.uic.edu/~radford/math516f06/WH4Sol.pdf.

    [32] R Raghavendran. Finite associative rings. Compositio Mathematica,
    21(2):195{229, 1969. Available at http://www.numdam.org/item?id=
    CM_1969__21_2_195_0.

    [33] Louis Halle Rowen et al. Ring theory, volume 1. Academic Press, 1988.

    [34] Yaghoub Shari . Lifting idempotents modulo nil ide-
    als. http://ysharifi.wordpress.com/2010/11/01/
    lifting-idempotents-modulo-nil-ideals/, Nov. 2010.

    [35] Timothy Wagner. Comaximal ideals in a commutative ring. http:
    //math.stackexchange.com/questions/10400, Nov. 2010.

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