强f-clean环，f-clean环的注记，拟clean环的扩张
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摘要
本硕士论文分为三部分:
第一部分:介绍clean环的研究概述及本文的主要工作。
第二部分:推广clean环的概念,提出了强f - clean环的概念,并且研究了强f - clean环上的一些性质,主要结果:
性质2.2.1.3强正则环是强f - clean环。
性质2.2.1.4如果R是强f - clean环,且幂等元是中心幂等元, V = R,对于任意的v∈V,存在w∈R,使v + w=0,则R通过V的平凡扩张T ( R, V)是强f - clean环。
性质2.2.1.5若交换环R是强f - clean环,则R是强f - clean环。
性质2.2.1.6如果R是任意交换环,则多项式环R [x ]不是强f - clean环。
性质2.2.1.7如果1 = e_1 +e_2++e_n ,n≥1,其中e_i是正交幂等元,且e_i Re_i是强f - clean环,则R是强f - clean环。
性质2.2.1.8设R是强f - clean环,并且e~2 =e∈R是中心幂等元,则e Re也是强f - clean环。
第三部分:对拟clean环的研究扩张。主要结果有:
性质3.5如果R是拟- clean环, W = R,对于任意的w∈R,存在t∈W,使t + w+wt+vt+wv′=0,则R通过W的理想扩张I ( R, W)是拟- clean环。
性质3.6 e~2 =e∈R,并且e是中心幂等元,如果a∈eRe在e Re中是拟- clean的,则a在R中也是拟- clean的。
We have three parts in this paper.
The first part: We introduce the grand results in theclean rings and my main work in the paper.
The second part: We generalize the concept ofclean rings ,and pose the concept of strongly f - cleanrings,and investigate some properties about strongly f - cleanrings.The following statement are the main results:
Propertion2.2.1.3 Strongly regular rings are strongly f - clean rings.
Propertion2.2.1.4 If R is strongly f - cleanrings ,and idempotents are center, V = R, -v∈V,- w∈R, make v + w=0,so the ordinary extention of T ( R, V) is strongly f - cleanrings.
Propertion2.2.1.5 If Abel rings R is f - clean rings ,so R is strongly f - clean rings.
Propertion2.2.1.6 If R is arbitrary Abel rings ,so polynomial rings is not strongly f - clean rings.
Propertion2.2.1.7 If 1 = e_1 +e_2++e_n ,n≥1,and ei are orthogonal idempotents, and ei Re iare strongly rings, so R are strongly f - clean rings.
Propertion2.2.1.8 Let R is strongly f - clean rings,and e~2 =e∈Ris center idempotents, e Reis also strongly f - clean rings.
The third part:We investigate some properties on quasi-clean rings.The following statement are the main results:
Propertion3.5 If R is quasi - clean rings, W = R, -w∈R, - t∈W,make t + w+wt+vt+wv′=0,so I ( R, V)is rings.
Propertion3.6 e~2 =e∈R,ande is center idempotents,if a∈eReis quasi - cleanin e Re,soa is quasi - cleanin R .
第一部分:介绍clean环的研究概述及本文的主要工作。
第二部分:推广clean环的概念,提出了强f - clean环的概念,并且研究了强f - clean环上的一些性质,主要结果:
性质2.2.1.3强正则环是强f - clean环。
性质2.2.1.4如果R是强f - clean环,且幂等元是中心幂等元, V = R,对于任意的v∈V,存在w∈R,使v + w=0,则R通过V的平凡扩张T ( R, V)是强f - clean环。
性质2.2.1.5若交换环R是强f - clean环,则R是强f - clean环。
性质2.2.1.6如果R是任意交换环,则多项式环R [x ]不是强f - clean环。
性质2.2.1.7如果1 = e_1 +e_2++e_n ,n≥1,其中e_i是正交幂等元,且e_i Re_i是强f - clean环,则R是强f - clean环。
性质2.2.1.8设R是强f - clean环,并且e~2 =e∈R是中心幂等元,则e Re也是强f - clean环。
第三部分:对拟clean环的研究扩张。主要结果有:
性质3.5如果R是拟- clean环, W = R,对于任意的w∈R,存在t∈W,使t + w+wt+vt+wv′=0,则R通过W的理想扩张I ( R, W)是拟- clean环。
性质3.6 e~2 =e∈R,并且e是中心幂等元,如果a∈eRe在e Re中是拟- clean的,则a在R中也是拟- clean的。
We have three parts in this paper.
The first part: We introduce the grand results in theclean rings and my main work in the paper.
The second part: We generalize the concept ofclean rings ,and pose the concept of strongly f - cleanrings,and investigate some properties about strongly f - cleanrings.The following statement are the main results:
Propertion2.2.1.3 Strongly regular rings are strongly f - clean rings.
Propertion2.2.1.4 If R is strongly f - cleanrings ,and idempotents are center, V = R, -v∈V,- w∈R, make v + w=0,so the ordinary extention of T ( R, V) is strongly f - cleanrings.
Propertion2.2.1.5 If Abel rings R is f - clean rings ,so R is strongly f - clean rings.
Propertion2.2.1.6 If R is arbitrary Abel rings ,so polynomial rings is not strongly f - clean rings.
Propertion2.2.1.7 If 1 = e_1 +e_2++e_n ,n≥1,and ei are orthogonal idempotents, and ei Re iare strongly rings, so R are strongly f - clean rings.
Propertion2.2.1.8 Let R is strongly f - clean rings,and e~2 =e∈Ris center idempotents, e Reis also strongly f - clean rings.
The third part:We investigate some properties on quasi-clean rings.The following statement are the main results:
Propertion3.5 If R is quasi - clean rings, W = R, -w∈R, - t∈W,make t + w+wt+vt+wv′=0,so I ( R, V)is rings.
Propertion3.6 e~2 =e∈R,ande is center idempotents,if a∈eReis quasi - cleanin e Re,soa is quasi - cleanin R .
引文
[1]Nicholson ,W. K .Lifting Idempotents and Exchange Rings . Trans. Amer. Mach. Soc, 1977,229:269-278
[2]Nicholson ,W. K I-rings, Trans.Amer. Math. Soc., 207(1975), 361-373
[3]Yu H. P. On quas-duo rings, Glasgow Math. J., 37(1995),21-31
[4]Chen H. Rings with many idempotents, Internat. J.Math. Sci., 22(3)(1999),547-558
[5]Searcoil M. Perturbation of linear operators by idempotents, Irish Math. Soc. Bull., 39(1997),10-13
[6]Nicholson W. K K. Varadarajan, Countable linear transformations are clean. Pro. Amer. Math. Soc.,126(1998),61-64
[7]Nicholson W. K. Strongly clean rings and Fitting's lemma, Comm. Alg., 27(1999),3583-3592
[8]Nicholson W. K. On exchange rings ,comm.Alg., 25(1997),1917-1918
[9]Anderson, V. P. Camillo, Commutative rings whose Elements are a sum of a unit and idempotent, Comm. Alg., 30(2002),3327-3336
[10]Nicholson W. K. Zhou Y Clean general rings, J. Algebra, 291(1)(2005),297-311
[11]Ye. Y Semiclean rings ,Comm. Alg., 31(2003),5609-5625
[12]Xiao G, Tong W, n-clean rings and weakly unit stable range rings ,Comm. Alg., 33(2005),1501-1517
[13]李炳君. clean环及其推广[D].国防科学技术大学.2008
[14]王尧,刘苏.拟clean环[J].辽宁师范大学学报(自然科学版)2007
[15]Camillo V P. Yu H P, Exchange rings ,Units and idempotents[J],Comm Algebra,1994,22(12):4737-4749
[16]Anderson F. W , Fuller K R. Rings and Categories of Modules [M]. GTM13. New York: Springer Verlang, 1991
[17]刘绍学.环与代数[M].北京:科学出版社,2001
[18]Nicholson W K. Lifting Idempotentsand Exchange Rings [J]. Trans Amer Math Soc,1977,229,269-278.
[19] Askar Tuganbaev. Rings Close to Regular [M]. Kluwer Academic Publishers, 2002.
[20]Chen Jianlong and Wang Zhou. When is 2×2 Matrix Ring over a commutative Local Ring Strongly Clean Matrix. J. Alg,2006,301:280-293
[21]Zhang Hongbo and Tong Weiting. Generalized Clean Rings. Journal of Nanjing University Mathematical Biquarterly,2005,22:183-188
[2]Nicholson ,W. K I-rings, Trans.Amer. Math. Soc., 207(1975), 361-373
[3]Yu H. P. On quas-duo rings, Glasgow Math. J., 37(1995),21-31
[4]Chen H. Rings with many idempotents, Internat. J.Math. Sci., 22(3)(1999),547-558
[5]Searcoil M. Perturbation of linear operators by idempotents, Irish Math. Soc. Bull., 39(1997),10-13
[6]Nicholson W. K K. Varadarajan, Countable linear transformations are clean. Pro. Amer. Math. Soc.,126(1998),61-64
[7]Nicholson W. K. Strongly clean rings and Fitting's lemma, Comm. Alg., 27(1999),3583-3592
[8]Nicholson W. K. On exchange rings ,comm.Alg., 25(1997),1917-1918
[9]Anderson, V. P. Camillo, Commutative rings whose Elements are a sum of a unit and idempotent, Comm. Alg., 30(2002),3327-3336
[10]Nicholson W. K. Zhou Y Clean general rings, J. Algebra, 291(1)(2005),297-311
[11]Ye. Y Semiclean rings ,Comm. Alg., 31(2003),5609-5625
[12]Xiao G, Tong W, n-clean rings and weakly unit stable range rings ,Comm. Alg., 33(2005),1501-1517
[13]李炳君. clean环及其推广[D].国防科学技术大学.2008
[14]王尧,刘苏.拟clean环[J].辽宁师范大学学报(自然科学版)2007
[15]Camillo V P. Yu H P, Exchange rings ,Units and idempotents[J],Comm Algebra,1994,22(12):4737-4749
[16]Anderson F. W , Fuller K R. Rings and Categories of Modules [M]. GTM13. New York: Springer Verlang, 1991
[17]刘绍学.环与代数[M].北京:科学出版社,2001
[18]Nicholson W K. Lifting Idempotentsand Exchange Rings [J]. Trans Amer Math Soc,1977,229,269-278.
[19] Askar Tuganbaev. Rings Close to Regular [M]. Kluwer Academic Publishers, 2002.
[20]Chen Jianlong and Wang Zhou. When is 2×2 Matrix Ring over a commutative Local Ring Strongly Clean Matrix. J. Alg,2006,301:280-293
[21]Zhang Hongbo and Tong Weiting. Generalized Clean Rings. Journal of Nanjing University Mathematical Biquarterly,2005,22:183-188