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数学进展 - 2020, Vol. 49(1): 29-38
研究论文
J-clean 环的推广
Generalizations of J-clean Rings

崔建, 秦龙
CUI Jian*, QIN Long

安徽师范大学数学系, 芜湖, 安徽, 241002
Department of Mathematics, Anhui Normal University, Wuhu, Anhui, 241002, P. R. China

收稿日期: 2018-12-17
出版日期: 2020-03-25
2020, Vol. 49(1): 29-38
DOI: 10.11845/sxjz.2018102b


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摘要 如果$R$中每个元素(对应地, 可逆元)均可表示为一个幂等元与环$R$的Jacobson根中一个元素之和, 则称环$R$是J-clean环(对应地, UJ环). 所有的J-clean环都是UJ环.作为UJ环的真推广, 本文引入GUJ环的概念, 研究GUJ环的基本性质和应用. 进一步地, 研究每个元素均可表示为一个幂等元与一个方幂属于环的Jacobson根的元素之和的环.
关键词 clean环J-clean环UJ环诣零clean环GJ-clean环    
Abstract:A ring $R$ is J-clean (respectively, UJ) if every element (respectively, unit) of $R$ is a sum of an idempotent and an element from the Jacobson radical of $R$. All J-clean rings are UJ rings. In this paper, we introduce the notion of a GUJ ring, which is a proper generalization of UJ rings. The properties and applications of GUJ rings are discussed. We also investigate rings for which every element is a sum of an idempotent and an element whose power belongs to the Jacobson radical.
Key wordsclean ring    J-clean ring    UJ ring    nil clean ring    GJ-clean ring
PACS:  O153.3  
通讯作者: E-mail: *cui368@ahnu.edu.cn   
[1] Abdolyousefi, M.S. and Chen, H.Y., Rings in which elements are sums of tripotents and nilpotents, J. Algebra Appl., 2018, 17(3): 1850042, 11 pp.
[2] Berberian S.K.,Baer $*$-rings, Grundlehren Math. Wiss., Vol. 195. Berlin: Springer-Verlag, 1972.
[3] Breaz S.,C$\check{\rm a}$lug$\check{\rm a}$reanu, G., Danchev, P. and Micu, T., Nil-clean matrix rings, Linear Algebra Appl., 2013, 439(10): 3115-3119.
[4] C$\check{\rm a}$lug$\check{\rm a}$reanu, G., UU rings, Carpathian J. Math., 2015, 31(2): 157-163.
[5] Chen H.Y.,Strongly J-clean rings, Comm. Algebra, 2010, 38(10): 3790-3804.
[6] Cui J.,Quasinilpotents in rings and their applications, Turkish J. Math., 2018, 42(5): 2854-2862.
[7] Danchev P.V.,Rings with Jacobson units, Toyama Math. [J]., 2016, 38: 61-74.
[8] Danchev P.V.,On exchange $\pi$-UU unital rings, Toyama Math. [J]., 2017, 39: 1-7.
[9] Danchev, P.V. and Lam, T.Y., Rings with unipotent units, Publ. Math. Debrecen, 2016, 88(3/4): 449-466.
[10] Diesl A.J.,Nil clean rings, [J]. Algebra, 2013, 383(1): 197-211.
[11] Han, J. and Nicholson, W.K., Extensions of clean rings, Comm. Algebra, 2001, 29(6): 2589-2595.
[12] Harte R.E.,On quasinilpotents in rings, Panam. Math. [J]., 1991, 1(1): 10-16.
[13] Karimi-Mansoub, A., Kosąn, M.T. and Zhou, Y.Q., Rings in which every unit is a sum of a nilpotent and an idempotent, In: Advances in Rings and Modules, AMS Contemporary Mathematics, Vol. 715, Providence. RI: AMS, 2018, 189-203.
[14] Kosąn M.T., Lee T.-K. and Zhou Y.Q., When is every matrix over a division ring a sum of an idempotent and a nilpotent? Linear Algebra Appl., 2014, 450: 7-12.
[15] Kosąn M.T., Leroy A. and Matczuk J., On UJ-rings, Comm. Algebra, 2018, 46(5): 2297-2303.
[16] Lam T.Y.,A First Course in Noncommutative Rings (Second Ed.), New York: Springer-Verlag, 2001.
[17] Levitzki J.,On the structure of algebraic algebras and related rings, Trans. Amer. Math. Soc., 1953, 74(3): 384-409.
[18] Matczuk, J., Conjugate (nil) clean rings and K$\ddot{\rm o}$the's problems, J. Algebra Appl., 2017, 16(4): 1750073, 14 pp.
[19] Nicholson W.K.,Lifting idempotents and exchange rings, Trans. Amer. Math. Soc., 1977, 229: 269-278.
[20] Nicholson W.K.,Strongly clean rings and Fitting's lemma, Comm. Algebra, 1999, 27(8): 3583-3592.
[21] Nicholson, W.K. and Zhou, Y.Q., Rings in which elements are uniquely the sum of an idempotent and a unit, Glasgow Math. [J]., 2004, 46(2): 227-236.
[22] Nicholson, W.K. and Zhou, Y.Q., Clean general rings, [J]. Algebra, 2005, 291(1): 297-311.
[23] $\check{\rm S}$ter, J., Rings in which nilpotents form a subring, Carpathian J. Math., 2016, 32(2): 251-258.
[24] Va${\rm \check{s}}$, L., $*$ -clean rings; some clean and almost clean Baer $*$-rings and von Neumann algebras, [J]. Algebra, 2010, 324(12): 3388-3400.
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