点态化完备代数正规类中的绝对半素代数类与绝对素代数类
The Absolutely Semiprime Algebras Class and the Absolutely Prime Algebras Class in Normal Classes of Complete Pointwise Algebras
摘要: 环及其它代数系统的根理论有了丰富的研究，Puczylowski建立了一般代数正规类的根理论。本文研究点态化完备代数正规类中的绝对半素代数类τs与绝对素代数类τ及其确定的上根性质Uτs，Uτ，证明了Uτs是超幂零根，Uτ是特殊根，τs是遗传根。
Abstract: The radicals of rings and other various algebraic structures have been researched very much. Puczylowski established the general theory of radicals of the objects called algebras. In this paper, we study the absolutely semiprime algebras class τs and the absolutely semiprime algebras class τ in the normal classes of pointwise complete algebras and the properties of the upper radical Uτs, Uτ determined by τs and τ, it is proved that Uτs is a supernilpotent radical, Uτ is a special radical and τs is a hereditary radical.
文章引用：杨宗文, 娄本功. 点态化完备代数正规类中的绝对半素代数类与绝对素代数类[J]. 理论数学, 2022, 12(11): 1934-1940. https://doi.org/10.12677/PM.2022.1211208

参考文献

[1] Száse, F.A. (1981) Radicals of Rings. John Wiley & Sons, New York.
[2] Gardner, B.J. and Wiegandt, R. (2004) Radical Theory of Rings. Marcel Dekker, Inc., New York and Basel. [Google Scholar] [CrossRef] 
[3] McDougall, R. (1999) A Generalisation of the Lower Radical Class. Bulletin of the Australian Mathematical Society, 59, 139-146. [Google Scholar] [CrossRef] 
[4] Van Leeuwen, L.C.A. and Heyman, G.A.P. (1975) A Radical Determined by a Class of Almost Nilpotent Rings. Acta Mathematica Academiae Scientiarum Hungaricae, 26, 259-262. [Google Scholar] [CrossRef] 
[5] Heyman, G.A.P, Jenkins, T.L. and Roux, H.J. (1982) Variations on Almost Nilpotent Rings, Their Radicals and Partitions. Acta Mathematica Academiae Scientiarum Hungaricae, 39, 11-15. [Google Scholar] [CrossRef] 
[6] Sands, A.D. (1985) On Almost Nilpotent Rings. Acta Mathematica Hungarica, 45, 41-43. [Google Scholar] [CrossRef] 
[7] Puczyłowski, E.R. (1986) A Note on Almost Nilpotent Rings. Acta Mathematica Hungarica, 48, 289-291. [Google Scholar] [CrossRef] 
[8] 梁治安. 关于几乎幂零环的一些结果[J]. 内蒙古大学学报(自然科学版), 1989, 20(4): 435-437.
[9] Heyman, G.A.P. (1990) On Almost Nilpotent Rings and Ideals. Acta Mathematica Hungarica, 56, 283-285. [Google Scholar] [CrossRef] 
[10] 张宪君. 关于绝对半素环和绝对半素根[J]. 纯粹数学与应用数学, 1993, 9(2): 57-60.
[11] 张宪君, 于淑兰. 绝对素环与绝对素根[J]. 黑龙江大学自然科学学报, 1998, 15(3): 21-22.
[12] Puczylowski, E.R. (1993) On General Theory of Radicals. Algebra Universalis, 39, 53-60. [Google Scholar] [CrossRef] 
[13] Wang, Y. and Zhang, A.H. (2002) Radicals and Semisimple Classes of the Class of Algebras. Journal of Anshan Normal University, 4, 5-10.
[14] 任艳丽, 王尧. 代数正规类中的遗传根与强半单根[J]. 数学研究与评论, 2004, 24(4): 597-602.
[15] Yang, Z.W. (2006) The Upper Radical Classes of the Class of Algebras. Journal Yunnan University (Natural Sciences Edition), 28, 8-11.
[16] Yang, Z.W. and Pan, J.M. (2008) The Supernilpotent Radical, Special Radical and Bear Radical in Normal Classes of Product Algebras. Southeast Asian Bulletin of Mathematics, 32, 181-192.
[17] Yang, Z.W. and Pan, J.M. (2010) The Radicals and Likemodules in Normal Classes of Complete Alagebras. Southeast Asian Bulletin of Mathematics, 34, 377-386.
[18] 杨宗文, 杨柱元. 完备代数正规类的根与右理想[J]. 昆明理工大学学报(理工版), 2006, 31(3): 112-116, 120.
[19] 杨宗文, 杨柱元. 子环的和与积[J]. 云南大学学报(自然科学版), 2007, 29(4): 335-338.
[20] 杨宗文, 杨柱元, 李友宝. 大半环子半环的和与积[J]. 昆明理工大学学报(理工版), 2007, 32(6): 113-118.
[21] 杨宗文, 杨柱元, 李友宝. 可积代数正规类中半素代数类及半素一致代数类确定的上根[J]. 数学理论与应用, 2008, 28(4): 71-75.
[22] Yang, Z.W., Yang, Z.Y. and Li, Y.B. (2010) The General Radicals Theory of the Big Semirings. Southeast Asian Bulletin of Mathematics, 34, 1149-1167.
[23] Yang, Z.W. and Yang, Z.Y. (2011) The Semihereditary and Semisupernilpotent Radicals in Normal Classes of Product Algebras. Southeast Asian Bulletin of Mathematics, 35, 891-902.
[24] 杨宗文, 何青海. 点态化完备代数正规类中的亚直既约代数类[J]. 理论数学, 2018, 8(5): 546-554. [Google Scholar] [CrossRef] 
[25] 杨宗文，何青海. 点态化完备代数正规类中的遗传幂等根、补根、对偶根、子幂等根及诣零根[J]. 理论数学, 2018, 8(6): 712-722. [Google Scholar] [CrossRef] 
[26] 杨宗文, 何青海. 点态化完备代数正规类中的λ-根和正则根[J]. 理论数学, 2019, 9(7): 836-842. [Google Scholar] [CrossRef] 
[27] 杨宗文, 何青海. 点态化完备代数正规类中的Jacobson代数和Boolean代数[J]. 理论数学, 2019, 9(9): 1009-1014. [Google Scholar] [CrossRef] 
[28] 杨宗文, 娄本功. 点态化完备代数正规类中Amitsur-Kurosh根的映射刻画[J]. 理论数学, 2020, 10(12): 1138-1144. [Google Scholar] [CrossRef] 
[29] 杨宗文, 娄本功. 点态化完备代数正规类中的低幂等根[J]. 理论数学, 2021, 11(1): 1-6. [Google Scholar] [CrossRef] 
[30] 杨宗文, 娄本功. 点态化完备代数正规类中的小理想[J]. 理论数学, 2021, 11(10): 1691-1695. [Google Scholar] [CrossRef] 
[31] 杨宗文, 娄本功. 完备代数正规类中的基根[J]. 理论数学, 2021, 11(12): 2012-2017. [Google Scholar] [CrossRef] 
[32] 杨宗文, 娄本功. 点态化完备代数正规类中的几乎幂零代数类[J]. 理论数学, 2022, 12(9): 1527-1535. [Google Scholar] [CrossRef] 

为你推荐