Exchange环与Clean环
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摘要
自从Ehrlich提出单位正则环的概念以来,国内外许多代数工作者对其进行了深入研究.随后,Warfield,Nicholson分别将单位正则环推广到exchange环,clean环.肖光世和佟文廷进一步将clean环推广到n - clean环.
本文主要讨论exchange环和clean环.首先,讨论clean环和2 - clean环的性质.其次,讨论n - clean一般环的幂级数环(包括左斜幂级数和右斜幂级数环)和矩阵环的性质,并且定义了强n - clean一般环,讨论强n - clean一般环的矩阵环和它的扩张.最后,讨论clean环,2 - clean环与exchange环关系.通过减弱“R是左拟- duo环”的条件,分别在“R是左(右)弱理想,广义弱理想,拟理想,补理想”的条件下来讨论clean环与exchange环的关系,得到了如果R是左(右)弱理想,广义弱理想,拟理想或补理想,那么R是clean环当且仅当R是exchange环.并将所得结果推广到2 - clean环.
Since the unit-regular ring conception was introduced by Ehrlich, many algebraic scholars have studied it. Later, it was extended to the exchange ring and the clean ring by Warfield and Nicholson. Later, the n-clean ring was introduced by Xiao Guangshi and Tong Wenting.
The main purpose of this paper is to discuss exchange rings and clean rings. Firstly, we introduced some properties of clean rings and 2-clean rings. Seconddly, we discussed n-clean power series rings (including left power series rings and right power series rings) and n-clean matrix rings. We also introduce strong n-clean general rings and some properties, such as the matrix and extention of strong n-clean rings. At the end of the paper, we study the relationship between clean rings, 2-clean rings and exchange rings. By weakening the condition that“R is a quai-duo ring”, some conditions are proved. If“every maximal left ideal of R is a GW-ideal”,“every maximal left ideal of R is a quasi-ideal”,“every maximal left ideal of R is a weakly right ideal”,“every complement left ideal of R is an ideal”, we obtain that R is exchange if and only if it is clean. We also extend the results to 2-clean rings.
本文主要讨论exchange环和clean环.首先,讨论clean环和2 - clean环的性质.其次,讨论n - clean一般环的幂级数环(包括左斜幂级数和右斜幂级数环)和矩阵环的性质,并且定义了强n - clean一般环,讨论强n - clean一般环的矩阵环和它的扩张.最后,讨论clean环,2 - clean环与exchange环关系.通过减弱“R是左拟- duo环”的条件,分别在“R是左(右)弱理想,广义弱理想,拟理想,补理想”的条件下来讨论clean环与exchange环的关系,得到了如果R是左(右)弱理想,广义弱理想,拟理想或补理想,那么R是clean环当且仅当R是exchange环.并将所得结果推广到2 - clean环.
Since the unit-regular ring conception was introduced by Ehrlich, many algebraic scholars have studied it. Later, it was extended to the exchange ring and the clean ring by Warfield and Nicholson. Later, the n-clean ring was introduced by Xiao Guangshi and Tong Wenting.
The main purpose of this paper is to discuss exchange rings and clean rings. Firstly, we introduced some properties of clean rings and 2-clean rings. Seconddly, we discussed n-clean power series rings (including left power series rings and right power series rings) and n-clean matrix rings. We also introduce strong n-clean general rings and some properties, such as the matrix and extention of strong n-clean rings. At the end of the paper, we study the relationship between clean rings, 2-clean rings and exchange rings. By weakening the condition that“R is a quai-duo ring”, some conditions are proved. If“every maximal left ideal of R is a GW-ideal”,“every maximal left ideal of R is a quasi-ideal”,“every maximal left ideal of R is a weakly right ideal”,“every complement left ideal of R is an ideal”, we obtain that R is exchange if and only if it is clean. We also extend the results to 2-clean rings.
引文
[1] Ehrlich G.. Unit-regular rings. Portugal. Math., 1968, 27: 209-212.
[2] Ehrlich G.. Units and one-sided units in regular rings. Trans. Amer. Math. Sco., 1976, 216: 81-90.
[3] Fuchs L. On a substitution property for modules. Monatshefte fur Math., 1971, 75: 198-204.
[4] Kaplansky I. Bass’s first stable range condition. Mimeographed notes, 1971.
[5] Henriksen M. On a class of regular rings that are elementary divisor rings. Arch. Der Math., 1973, 24: 133-141.
[6] Crawley P, Jonsson B. Refinements for infinite direct decompositions of algebraic systems. Pacific J. Math., 1964, 14: 797-855.
[7] Warfield R B. A Kull-Schmidt theorem for infinite sums of modules. Proc. Amer. Math. Soc., 1969, 22: 460-465.
[8] Warfield R B. Exchange rings and decompositions of modules. Math.Ann., 1972, 199: 31-36.
[9] Nicholson W K. Lifting idempotents and exchange rings. Trans. Amer. Math. Soc., 1977, 229: 269-278.
[10] Camillo V P, Yu H P. Exchange Rings, Units and Idempotents. Comm. Algebra, 1994, 22: 4737-4749.
[11] Yu H P. On quasi-duo rings, Glasses of modules with the exchange property. J. Algebra, 1984, 88: 416-434.
[12] Chen H. Exchange rings with artinian primitive factors. Algebra Represent Theory, 1999, 2: 201-207.
[13] Birkenmeier G F. Idempotent sang completely semiprime ideals. Comm. Algebra, 1983, 11: 567-580.
[14] Lam T Y. A First Course in Noncommutative Rings. Springer-verlag, Berlin-Heidelberg-New York, 2001.
[15] Song X M, Yin X B. Generalizations of V-rings. Kyungpook Math.J., 2005, 45: 357-362.
[16] Zhang J L. Characterizations of strongly regular rings. Northeast Math. J., 1994, 3: 359-364.
[17]宋贤梅. V-环的推广.安徽师范大学硕士学位论文,安徽芜湖,安徽师范大学, 2002.
[18] Menal P. On subgroups of Gl 2over Banach algebras and von Neumann regular rings which are nomalized by elementary matrices. J. Algebra, 1991, 138: 99-120.
[19] Xiao G S, Tong W T. N-Clean rings and weakly unit stable range rings. Comm. Algebra, 2005,33: 1501-1517.
[20] Nichloson W K, Zhou Y. Clean general rings. J. Algebra, 2005, 291: 297-311.
[21] Xiao G S, Tong W T. N-Clean rings. Algebra Colloquium, 2006, 13: 599-606.
[22] Han J, Nicholson W K. Extensions of clean rings. Comm.Algebra, 2001, 29: 2589-2595.
[23] Chen W X. P-clean. International Journal of Mathematics and Mathematical Sciences, 2006, 2006: 1-7.
[24] Anderson D D, Camillo V P. Commutative rings whose elements are a sum of a unit and idempotent. Comm. Algebra, 2002, 30: 3327-3336.
[25] McGobern W Wm. A characterization of commutative clean rings. Int. J. Math. Game Theory Algebra, 2006, 15: 403-413.
[26] Nicholson W K. Strongly clean rings and Fitting’s lemma. Comm. Algebra, 1999, 27: 3583-3592.
[27] Nicholson W K, Zhou Y. Clean general rings. J.Algebra, 2005, 291: 297-311.
[28] Ara P. Extensions of exchange rings. J.Algebra, 1997, 197: 409-423.
[29]崔书英,陈卫星. Clean一般环的几个结果.大学数学, 2008, 24: 66-69.
[30]王静. N-clean一般环.甘肃科学学报, 2007, 19: 26-28.
[31]姜侠.一般clean环的扩张.兰州交通大学学报, 2006, 25: 149-150.
[32] Wang Z, Chen J L. On strongly clean general rings. Journal of Mathematical Research and Exposition, 2007, 27: 28-34.
[33]周海燕,王小东. Von Neumann正则环与左SF-环.数学研究与评论, 2004, 24: 679-683.
[34] Xiao G S. On strongly regular rings and SF-rings. F. J. M. S., 2009, 32: 387-396.
[35] Xiao G S, Tong W T. Rings whose every simple left R-modules is GP-injective. Southeast Asian Bulletin Math., 2006, 5: 969-980.
[36] Ramamurthi V S. On the injectivity and flatness of certain cyclic modules. Proc. Ammer. Math. Soc., 1975, 48: 21-25.
[37] Goodearl K R. Von Neumann Regular Rings. Kriegex Publishing Company, Florida, 1999.
[2] Ehrlich G.. Units and one-sided units in regular rings. Trans. Amer. Math. Sco., 1976, 216: 81-90.
[3] Fuchs L. On a substitution property for modules. Monatshefte fur Math., 1971, 75: 198-204.
[4] Kaplansky I. Bass’s first stable range condition. Mimeographed notes, 1971.
[5] Henriksen M. On a class of regular rings that are elementary divisor rings. Arch. Der Math., 1973, 24: 133-141.
[6] Crawley P, Jonsson B. Refinements for infinite direct decompositions of algebraic systems. Pacific J. Math., 1964, 14: 797-855.
[7] Warfield R B. A Kull-Schmidt theorem for infinite sums of modules. Proc. Amer. Math. Soc., 1969, 22: 460-465.
[8] Warfield R B. Exchange rings and decompositions of modules. Math.Ann., 1972, 199: 31-36.
[9] Nicholson W K. Lifting idempotents and exchange rings. Trans. Amer. Math. Soc., 1977, 229: 269-278.
[10] Camillo V P, Yu H P. Exchange Rings, Units and Idempotents. Comm. Algebra, 1994, 22: 4737-4749.
[11] Yu H P. On quasi-duo rings, Glasses of modules with the exchange property. J. Algebra, 1984, 88: 416-434.
[12] Chen H. Exchange rings with artinian primitive factors. Algebra Represent Theory, 1999, 2: 201-207.
[13] Birkenmeier G F. Idempotent sang completely semiprime ideals. Comm. Algebra, 1983, 11: 567-580.
[14] Lam T Y. A First Course in Noncommutative Rings. Springer-verlag, Berlin-Heidelberg-New York, 2001.
[15] Song X M, Yin X B. Generalizations of V-rings. Kyungpook Math.J., 2005, 45: 357-362.
[16] Zhang J L. Characterizations of strongly regular rings. Northeast Math. J., 1994, 3: 359-364.
[17]宋贤梅. V-环的推广.安徽师范大学硕士学位论文,安徽芜湖,安徽师范大学, 2002.
[18] Menal P. On subgroups of Gl 2over Banach algebras and von Neumann regular rings which are nomalized by elementary matrices. J. Algebra, 1991, 138: 99-120.
[19] Xiao G S, Tong W T. N-Clean rings and weakly unit stable range rings. Comm. Algebra, 2005,33: 1501-1517.
[20] Nichloson W K, Zhou Y. Clean general rings. J. Algebra, 2005, 291: 297-311.
[21] Xiao G S, Tong W T. N-Clean rings. Algebra Colloquium, 2006, 13: 599-606.
[22] Han J, Nicholson W K. Extensions of clean rings. Comm.Algebra, 2001, 29: 2589-2595.
[23] Chen W X. P-clean. International Journal of Mathematics and Mathematical Sciences, 2006, 2006: 1-7.
[24] Anderson D D, Camillo V P. Commutative rings whose elements are a sum of a unit and idempotent. Comm. Algebra, 2002, 30: 3327-3336.
[25] McGobern W Wm. A characterization of commutative clean rings. Int. J. Math. Game Theory Algebra, 2006, 15: 403-413.
[26] Nicholson W K. Strongly clean rings and Fitting’s lemma. Comm. Algebra, 1999, 27: 3583-3592.
[27] Nicholson W K, Zhou Y. Clean general rings. J.Algebra, 2005, 291: 297-311.
[28] Ara P. Extensions of exchange rings. J.Algebra, 1997, 197: 409-423.
[29]崔书英,陈卫星. Clean一般环的几个结果.大学数学, 2008, 24: 66-69.
[30]王静. N-clean一般环.甘肃科学学报, 2007, 19: 26-28.
[31]姜侠.一般clean环的扩张.兰州交通大学学报, 2006, 25: 149-150.
[32] Wang Z, Chen J L. On strongly clean general rings. Journal of Mathematical Research and Exposition, 2007, 27: 28-34.
[33]周海燕,王小东. Von Neumann正则环与左SF-环.数学研究与评论, 2004, 24: 679-683.
[34] Xiao G S. On strongly regular rings and SF-rings. F. J. M. S., 2009, 32: 387-396.
[35] Xiao G S, Tong W T. Rings whose every simple left R-modules is GP-injective. Southeast Asian Bulletin Math., 2006, 5: 969-980.
[36] Ramamurthi V S. On the injectivity and flatness of certain cyclic modules. Proc. Ammer. Math. Soc., 1975, 48: 21-25.
[37] Goodearl K R. Von Neumann Regular Rings. Kriegex Publishing Company, Florida, 1999.