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系統識別號 U0026-1507202010275500
論文名稱(中文) 近環之原始性與單純性
論文名稱(英文) Primitivity and Simplicity in Nearrings
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 108
學期 2
出版年 109
研究生(中文) 丁峻威
研究生(英文) Chun-Wei Ting
學號 L16071108
學位類別 碩士
語文別 英文
論文頁數 23頁
口試委員 指導教授-柯文峰
口試委員-黃世昌
口試委員-章源慶
中文關鍵字 近環  原始環 
英文關鍵字 Nearring  Primitive ring 
學科別分類
中文摘要 最小理想與最小左理想的概念在環論與近環裡扮演著重要的角色,為了研究更深層的結果,會需要使用雅各布森根基的概念。在這篇文章裡,我們會探討近環的根基與其衍伸出來好的結果,我們還會給出非交換環論裡的一些結果與證明。
英文摘要 The concept of minimal ideals and minimal left ideals plays a dominant
role in ring and nearring theory. To study deeper results on this
concept demands some powerful tools. Therefore, the idea of the radical
in ring and nearring theory is essential. In ring theory, one of the
importance of the Jacobson radical J(R) lies in the fact that every
nil left ideal and every nil right ideal in R is contained in J(R).

Nearrings is a generalization of rings which arise naturally from
mappings on groups. Since every ring is a nearring, the true statements
in nearrings are useful in ring theory. In this article, the radicals
for nearrings and their good properties will be discussed. Moreover,
some results in noncommutative ring theory will be proved in this
article.
論文目次 Fundamental definitions and properties--------1
Modularity and Quasiregularity--------4
Nilness and Nilpotency--------5
Radicals for Nearrings--------6
Some results in Nearrings--------12
Some results in Rings --------21
References--------22
參考文獻 [1] V. A. Andrunakievicˇ, Radicals of associative rings. I, American Mathematical Society Translations: Series 2, 52 (1966), 95–128.
[2] V. A. Andrunakievicˇ, Radicals of associative rings. II, American Mathematical Society Translations: Series 2, 52 (1966), 129–149.
[3] G. Betsch, Struktursa ̈tze fu ̈r Fastringe, Diss., University of Tu ̈bingen, 1963.
[4] R. Brauer, On the nilpotency of the radical of a ring, Bull. Amer. Math. Soc.48(1942), 752-758.
[5] N. J. Divinsky, Rings and Radicals, Mathematical Expositions 14 ,1965.
[6] C. C. Ferrero and G. Ferrero, Nearrings: Some Developments Linked to Semigroups and Groups, Advances in Math. 4, 2002.
[7] M. Ferrero and E. R. Puczylowski, On rings which are sums of two subrings, Arch. Math. 53 (1989), 4–10.
[8] I. N. Herstein, Noncommutative Rings, Carus Math. Monographs 15, 1968.
[9] T. W. Hungerford, Algebra, Graduate Texts in Mathematics 73, SpringerVerlag,1980.
[10] N. Jacobson, Structure of Rings, American Mathematical Soc. Colloquium Pub. 37, 1956.
[11] J. P. Jans, Projective injective modules, Pacific Journal of Mathematics 9(1959),1103-1108.
[12] K. Kaarli, Survey on the radical theory of near-rings, Contributions to GeneralAlgebra 4 (1985), 45–62.
[13] I. Kaplansky, Fields and Rings, Chicago Lectures in Mathematics, 1969 (2nd ed.1972).
[14] N. H. McCoy, The Theory of Rings, Chelsea Pub. Co., 1973.
[15] W. K. Nicholson, A short proof of the Wedderburn-Artin theorem, New Zealand Journal of Math. 22 (1993), 83–86.
[16] G. Pilz, Near-rings: What they are and what they are good for, Contemporary Mathematics 9 (1982), 97–119.
[17] G. Pilz, Near-rings: The Theory and its Applications, North-Holland Math. Studies 23, 1977 (Revised ed. 1983).
[18] D. Ramakotaiah, Radicals for near-rings, Math. Zeitschr. 97 (1967), 45–56.
[19] D. Ramakotaiah, Structure of -primitive near-rings, Math. Zeitschr. 110(1969),15–26.
[20] D. Ramakotaiah, A radical for near-rings, Arch. Math. 23 (1972), 482–483.
[21] J. C. Robson, Do simple rings have unity elements? , Journal of Algebra 7 (1967), 140–143.
[22] A. Smoktunowicz, On some results related to Ko ̈the’s conjecture, Serdica Math. J. 27 (2001), 159–170.
[23] A. Smoktunowicz, Some results in noncommutative ring theory, Proceedings of the International Congress of Mathematicians 2 (2006), 259–269.
[24] G. Wendt, Minimal ideals and primitivity in near-rings, Taiwanese Journal of Mathematics 23 (2019), 799–820.
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