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 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。
 The concept of minimal ideals and minimal left ideals plays a dominantrole in ring and nearring theory. To study deeper results on thisconcept demands some powerful tools. Therefore, the idea of the radicalin ring and nearring theory is essential. In ring theory, one of theimportance of the Jacobson radical J(R) lies in the fact that everynil left ideal and every nil right ideal in R is contained in J(R).Nearrings is a generalization of rings which arise naturally frommappings on groups. Since every ring is a nearring, the true statementsin nearrings are useful in ring theory. In this article, the radicalsfor nearrings and their good properties will be discussed. Moreover,some results in noncommutative ring theory will be proved in thisarticle.
 Fundamental definitions and properties--------1Modularity and Quasiregularity--------4Nilness and Nilpotency--------5Radicals for Nearrings--------6Some results in Nearrings--------12Some results in Rings --------21References--------22
 [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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