拟Clean环与强拟Armendariz环
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摘要
本硕士论文分为三部分。
第一部分：介绍Clean环和Armendariz环的研究概述以及本文的主要工作。
第二部分：我们推广Clean环的概念，提出了拟Clean环的概念，并且研究了拟Clean环上的一些性质。主要结果：
定理2．2．1．2：设e~2=e∈R。如果eRe和(1-e)R(1-e)都是拟Clean环，则R是拟Clean环。
定理2．2．1．4：若R是拟Clean环，则矩阵环M_n(R)是拟Clean环。
定理2．2．2．1：设ψ=0，φ=0，则有
(1) C=(?)是拟Clean环当且仅当A，B都是拟Clean环。
(2) C=(?)是G-Clean环当且仅当A，B都是G-Clean环。
定理2．2．3．2幂级数环R[[x]]是拟Clean环当且仅当R亦然。
定理2．2．3．5如果R是拟Clean环，幂等元模J可提升，且R只有正交幂等元的有限集，则
(1) R是Semiperfect环。(2)R是Clean环。(3)R是Exchange环。
第三部分：我们给出强拟Armendariz环的概念，研究强Armen-dariz环和强拟Armendariz环上的一些性质。主要结果：
定理3．2．2．1设R是强Armendariz环，则R在右零化子上满足升链条件当且仅当R[[x]]亦然。
定理3．2．2．3如果R是强Armendariz环，则R是拟Baer环当且仅当R[[x]]是拟Baer环。
定理3．2．4．4如果R是强拟Armendariz环，且R Morita等价于S，则S是强拟Armendariz环。
定理3．2．4．5如果R是强拟Armendariz环，则对于每个正整数n，都有M_n(R)也是强拟Armendariz环。
We have three parts in this paper.
The first part:We introduce the grand results in the Clean ringand Armendariz ring, and our main work in the paper.
The second part:We generalize the concept of Clean rings, andpose the concept of Quasi-Clean rings, and investigate some propertiesabout Quasi-Clean rings. The following statement are the main results:
Theorem 2.2.1.2 Let e~2=e∈R be such that ere and (1-e)R(1-e) are both Quasi-Clean rings. Then R is a Quasi-Cleanrings.
Theorem 2.2.1.4 If R is a Quasi-Clean ring,then Matrix ringM_n(R) is a Quasi-Clean ring.
Theorem 2.2.2.1 Letψ=0,φ=0, then we have
(1) C=(?)is a Quasi-Clean ring if and only if A, B are bothQuasi-Clean rings.
(2)C=(?) is a G-Clean ring if and only if A, B are bothG-Clean rings.
Theorem 2.2.3.2 The ring R[[x]] is a Quasi-Clean ring if andonly if R is a Quasi-Clean rings.
Theorem 2.2.3.5 If R is a拟Clean ring, idempotents lift stronglymodulo J, and R contains no infinite set of orthogonal idempotents, thefollowing are equivalent:
(1) R is a Semiperfect ring. (2)R is a Clean ring. (3)R is a Exchangering.
The third part: we pose the concept of strongly Quasi-ArmendarizRings, and investigate some properties on strongly Armendariz Ringsand strongly Quasi-Armendariz Rings.
Theorem 3.2.2.1 If R is a strongly Armendariz ring,then R sat- isfies the ascending chain condition on right annihilator if and only ifso is R[[x]].
Theorem 3.2.2.3 If R is a strongly Armendariz ring, then R is aQuai-Bare ring if and only if R[[x]] is a Quai-Bare ring.
Theorem 3.2.4.4 If R is a strongly Quasi-Armendariz ring and ifR Morita equivalent to a ring S, then S is a strongly Quasi-Armendarizring.
Theorem 3.2.4.5 If R is a strongly Quasi- Armendariz ring,then,forany positive integer n, M_n(R) is a strongly Quasi- Armendariz ring.
第一部分：介绍Clean环和Armendariz环的研究概述以及本文的主要工作。
第二部分：我们推广Clean环的概念，提出了拟Clean环的概念，并且研究了拟Clean环上的一些性质。主要结果：
定理2．2．1．2：设e~2=e∈R。如果eRe和(1-e)R(1-e)都是拟Clean环，则R是拟Clean环。
定理2．2．1．4：若R是拟Clean环，则矩阵环M_n(R)是拟Clean环。
定理2．2．2．1：设ψ=0，φ=0，则有
(1) C=(?)是拟Clean环当且仅当A，B都是拟Clean环。
(2) C=(?)是G-Clean环当且仅当A，B都是G-Clean环。
定理2．2．3．2幂级数环R[[x]]是拟Clean环当且仅当R亦然。
定理2．2．3．5如果R是拟Clean环，幂等元模J可提升，且R只有正交幂等元的有限集，则
(1) R是Semiperfect环。(2)R是Clean环。(3)R是Exchange环。
第三部分：我们给出强拟Armendariz环的概念，研究强Armen-dariz环和强拟Armendariz环上的一些性质。主要结果：
定理3．2．2．1设R是强Armendariz环，则R在右零化子上满足升链条件当且仅当R[[x]]亦然。
定理3．2．2．3如果R是强Armendariz环，则R是拟Baer环当且仅当R[[x]]是拟Baer环。
定理3．2．4．4如果R是强拟Armendariz环，且R Morita等价于S，则S是强拟Armendariz环。
定理3．2．4．5如果R是强拟Armendariz环，则对于每个正整数n，都有M_n(R)也是强拟Armendariz环。
We have three parts in this paper.
The first part:We introduce the grand results in the Clean ringand Armendariz ring, and our main work in the paper.
The second part:We generalize the concept of Clean rings, andpose the concept of Quasi-Clean rings, and investigate some propertiesabout Quasi-Clean rings. The following statement are the main results:
Theorem 2.2.1.2 Let e~2=e∈R be such that ere and (1-e)R(1-e) are both Quasi-Clean rings. Then R is a Quasi-Cleanrings.
Theorem 2.2.1.4 If R is a Quasi-Clean ring,then Matrix ringM_n(R) is a Quasi-Clean ring.
Theorem 2.2.2.1 Letψ=0,φ=0, then we have
(1) C=(?)is a Quasi-Clean ring if and only if A, B are bothQuasi-Clean rings.
(2)C=(?) is a G-Clean ring if and only if A, B are bothG-Clean rings.
Theorem 2.2.3.2 The ring R[[x]] is a Quasi-Clean ring if andonly if R is a Quasi-Clean rings.
Theorem 2.2.3.5 If R is a拟Clean ring, idempotents lift stronglymodulo J, and R contains no infinite set of orthogonal idempotents, thefollowing are equivalent:
(1) R is a Semiperfect ring. (2)R is a Clean ring. (3)R is a Exchangering.
The third part: we pose the concept of strongly Quasi-ArmendarizRings, and investigate some properties on strongly Armendariz Ringsand strongly Quasi-Armendariz Rings.
Theorem 3.2.2.1 If R is a strongly Armendariz ring,then R sat- isfies the ascending chain condition on right annihilator if and only ifso is R[[x]].
Theorem 3.2.2.3 If R is a strongly Armendariz ring, then R is aQuai-Bare ring if and only if R[[x]] is a Quai-Bare ring.
Theorem 3.2.4.4 If R is a strongly Quasi-Armendariz ring and ifR Morita equivalent to a ring S, then S is a strongly Quasi-Armendarizring.
Theorem 3.2.4.5 If R is a strongly Quasi- Armendariz ring,then,forany positive integer n, M_n(R) is a strongly Quasi- Armendariz ring.
引文
[1] Nicholson,W.K. Lifting Idempotents and Exchange Rings. Trans. Amer. Math. Soc,1977, 229: 269-278.
[2] Camillo,V.P and Yu Huaping.Exchange Rings,Units and Idempotents. Comm. Agl.1994,35(2):4737-4749.
[3] Han.J and Nicholson.W.K. Extensions of Clean Rings. Comm. Algebra.2001,29(6):2579-2595.
[4] Nicholson.W.K and Varadarajan.K Countable Linear Transformation are Clean.Pro. Amer. Math. Soc, 1998126 : 61-64.
[5] Nicholson.W.K and Varadarajan.K.Z. Clean Endomorphism Rings. Arch. Math.(Basel). 2001,26(6):340-343.
[6] Khurana.D and Lam.T.Y. Clean Matrices and Unit-Regular Matrices.J.Alg,2004,280:683-698.
[7] Camillo.V.P and Khurana.D. A Characterization of Unit Regular Rings. Comm.Algebra, 2001,29(5):2293-2295.
[8] Nicholson.W.K. Strongly Clean Rings and Lifting's Lemma. Comm. Algebra,1999,27(8):3583-3592.
[9] Chen Jianlong and Zhou Yang. On Strongly Clean Matrix and Triangular Ma-trix.J.Alg, 2005,28:686-698.
[10] Sanchez C E. On Strongly Clean Rings,unpublished.
[11] Chen Jianlong and Yang X. On two open problem about Strongly Clean Rings.Bull. Austral. Math. Sco, 2004,301:279-282.
[12] Chen Jianlong and Wang Zhou. When is 2 x 2 Matrix Ring over a commutative Local Ring Strongly Clean Matrix. J. Alg, 2006,301:280-293.
[13] Ye Yunqing. Semiclean Rings. Comm. Algebra, 2001,31(ll):5609-5625.
[14] Zhang Hongbo and Tong Weiting. Generalized Clean Rings. Journal of Nanjing University Mathematical Biquarterly, 2005,22: 183-188.
[15] Rege.M.B and Chhawchharia.S. Armendariz Rings. Proc. Japan. Acad. Ser.A Math.Sci,1997, 73: 14-17.
[16] Armendariz.E.P. A Note on Bare and PP Rings. J. Austral. Math. Sco,1974, 18:470-473.
[17] Anderson.D.D and Victor C. Armendariz Rings and Gaussian Rings. Comm. Algebra, 1998,26(7):2265-2272.
[18] Kim.N.K and Lee.Y.Armendariz Rings and Reduced Rings.J.Algebra,2000,223(6):477-488.
[19] Lee.T.K and Yiqing Zhou. Armendariz Rings and Reduced Rings. Comm.Algebra,2004,32(6):2287-2299.
[20] Lee.T.K and Wong.T.L.On Armendariz Rings.Houston.J.Algebra, 2003,29(3):583-593.
[21] 王尧,景立敏.环的 Armendariz 性.待发表.
[22] Hong. C. Y and Kim. N. Y. On Skew Armendariz Rings. Comm. Algebra, 2003, 31(1): 103-122.
[23] Weixing Chen and Wengting Tong. A Note on Skew Armendariz Rings. Comm. Algebra, 2005, 33:1137-1140.
[24] 郭颖，杜现昆，谢敬然．Armendariz环和斜Armendariz环．吉林大学学报(理学版)，2005，43(3)：253-257．
[25] Huh.C, Lee.K and Smoktunowicz.A. Armendariz Rings and Semicommutative Rings. Comm. Algebra, 2002, 30(2):751-761.
[26] Zhang Mianmian. Some Properties on Baer PP an PS Rings. 数学研究与评论．2006, 26: 685-693.
[27] Yasuyuki Hirano. On Annihilator Ideals of a Polynomial Ring over a Noncommutative Ring. J. Pure Algebra, 2002, 168:45-52.
[28] 刘绍学．环与代数．北京：科学出版社，2001．
[29] 王尧，任艳丽．具有一对零同态的Morita Context环．吉林大学学报(理学版)，2006，44(3)：310-324．
[30] Hungerford. T. W. Algebra. GTM73, New York: Springer-Verlang, 1974.
[31] Anderson. F. W and Fuller. K. R. Rings and Categories of Modules. GTM13. New York: Springer-Verlang, 1991. 1997, 73: 14-17.
[32] Hong. C. Y and Kim. T. Y. Ore extension of Baer and PP ring. J. Pure Algebra, 2000, 151: 215-226.
[2] Camillo,V.P and Yu Huaping.Exchange Rings,Units and Idempotents. Comm. Agl.1994,35(2):4737-4749.
[3] Han.J and Nicholson.W.K. Extensions of Clean Rings. Comm. Algebra.2001,29(6):2579-2595.
[4] Nicholson.W.K and Varadarajan.K Countable Linear Transformation are Clean.Pro. Amer. Math. Soc, 1998126 : 61-64.
[5] Nicholson.W.K and Varadarajan.K.Z. Clean Endomorphism Rings. Arch. Math.(Basel). 2001,26(6):340-343.
[6] Khurana.D and Lam.T.Y. Clean Matrices and Unit-Regular Matrices.J.Alg,2004,280:683-698.
[7] Camillo.V.P and Khurana.D. A Characterization of Unit Regular Rings. Comm.Algebra, 2001,29(5):2293-2295.
[8] Nicholson.W.K. Strongly Clean Rings and Lifting's Lemma. Comm. Algebra,1999,27(8):3583-3592.
[9] Chen Jianlong and Zhou Yang. On Strongly Clean Matrix and Triangular Ma-trix.J.Alg, 2005,28:686-698.
[10] Sanchez C E. On Strongly Clean Rings,unpublished.
[11] Chen Jianlong and Yang X. On two open problem about Strongly Clean Rings.Bull. Austral. Math. Sco, 2004,301:279-282.
[12] Chen Jianlong and Wang Zhou. When is 2 x 2 Matrix Ring over a commutative Local Ring Strongly Clean Matrix. J. Alg, 2006,301:280-293.
[13] Ye Yunqing. Semiclean Rings. Comm. Algebra, 2001,31(ll):5609-5625.
[14] Zhang Hongbo and Tong Weiting. Generalized Clean Rings. Journal of Nanjing University Mathematical Biquarterly, 2005,22: 183-188.
[15] Rege.M.B and Chhawchharia.S. Armendariz Rings. Proc. Japan. Acad. Ser.A Math.Sci,1997, 73: 14-17.
[16] Armendariz.E.P. A Note on Bare and PP Rings. J. Austral. Math. Sco,1974, 18:470-473.
[17] Anderson.D.D and Victor C. Armendariz Rings and Gaussian Rings. Comm. Algebra, 1998,26(7):2265-2272.
[18] Kim.N.K and Lee.Y.Armendariz Rings and Reduced Rings.J.Algebra,2000,223(6):477-488.
[19] Lee.T.K and Yiqing Zhou. Armendariz Rings and Reduced Rings. Comm.Algebra,2004,32(6):2287-2299.
[20] Lee.T.K and Wong.T.L.On Armendariz Rings.Houston.J.Algebra, 2003,29(3):583-593.
[21] 王尧,景立敏.环的 Armendariz 性.待发表.
[22] Hong. C. Y and Kim. N. Y. On Skew Armendariz Rings. Comm. Algebra, 2003, 31(1): 103-122.
[23] Weixing Chen and Wengting Tong. A Note on Skew Armendariz Rings. Comm. Algebra, 2005, 33:1137-1140.
[24] 郭颖，杜现昆，谢敬然．Armendariz环和斜Armendariz环．吉林大学学报(理学版)，2005，43(3)：253-257．
[25] Huh.C, Lee.K and Smoktunowicz.A. Armendariz Rings and Semicommutative Rings. Comm. Algebra, 2002, 30(2):751-761.
[26] Zhang Mianmian. Some Properties on Baer PP an PS Rings. 数学研究与评论．2006, 26: 685-693.
[27] Yasuyuki Hirano. On Annihilator Ideals of a Polynomial Ring over a Noncommutative Ring. J. Pure Algebra, 2002, 168:45-52.
[28] 刘绍学．环与代数．北京：科学出版社，2001．
[29] 王尧，任艳丽．具有一对零同态的Morita Context环．吉林大学学报(理学版)，2006，44(3)：310-324．
[30] Hungerford. T. W. Algebra. GTM73, New York: Springer-Verlang, 1974.
[31] Anderson. F. W and Fuller. K. R. Rings and Categories of Modules. GTM13. New York: Springer-Verlang, 1991. 1997, 73: 14-17.
[32] Hong. C. Y and Kim. T. Y. Ore extension of Baer and PP ring. J. Pure Algebra, 2000, 151: 215-226.