Clean环的推广
|
摘要
自1977年，Nicholson提出clean环，国内外很多代数学家对其进行了深入的研究，且作为单位正则环的推广，clean环有广泛的实际应用．继clean环后，Xiao和Zhang将clean环推广，分别提出了n-clean环和广义clean环．
本文第二部分首先将n-clean环和广义clean环进行推广，引入一类新的环----广义n-clean环．如果(?)_x∈R，x =w+u_1+u_2+...+u_n，其中w是单位正则的，u_1,u_2,...,u_n∈U(R)．显然n-clean环和广义clean环一定是广义n-clean环，且举例说明广义n-clean环不一定是广义clean环．证明了幂等元是中心的，广义n-clean环是(n+1)-clean环．并讨论了广义n-clean的基本性质，以及与n-good环之间的关系．其次，主要讨论广义n-clean环的矩阵扩张．我们知道，任意环上的矩阵环M_m(R)是3-clean环，而文献[20]中证明了行列有限的矩阵环是2-clean环，因此，任意环R上的m×m矩阵环M_m(R)是2-clean的，本文采取对矩阵进行分解的方法，给出另外一种证明．最后，主要讨论了广义n-clean环的多项式扩张．证明环R是2-素环，多项式环R[x]不是广义n-clean环．
本文第三部分，主要是将半-clean环推广到一般环上，定义了半-clean一般环．称一般环I为半-clean的，如果(?)_x∈I，x=a+q，其中a是周期的，即对于某正整数k和l(k≠l)有a~k = a~l，a∈I，且q∈Q．最后论及研究了半-clean一般环的性质，并证明半-clean一般环上的矩阵环还是半-clean一般环．
In 1977,Nicholson proposed clean rings.A lot of algebra scholars at home and abroad begin to research clean rings.And as the generalization of unit regular rings,clean rings have practical application.Following clean rings,Xiao and Zhang generalized clean rings to n-clean rings and generalized clean rings.
In section two,firstly the notion of generalized n-clean rings is introduced.These rings are shown to be a natual generalization of n-clean rings and generalized clean rings.A ring R is called generalized n-clean if (?)_x∈R，x =w+u_1+u_2+...+u_n ,where w is a unit regular element and u_1,u_2,...,u_n∈U(R).Clearly n-clean rings and generalized clean rings must be generalized n-clean rings.And an example that generalized n-clean rings need not be generalized clean rings will be given.We will prove that if idempotents are central generalized n-clean rings are (n+1)-clean rings.The basic properties of generalized n-clean rings and the relationship between generalize n-clean rings and n-good rings will be discussed.Secondly,we will discuss matrix extensions of generalized n-clean rings.It is known that any matrix ring is 3-clean.And in [20],it is proved that the row and column-finite matrix ring is 2-clean.Thus any matrix ring is 2-clean.We will decompose the matrix and give another proof.Finally,we will mainly discuss polynomial extensions of generalized n-clean rings.It will be proved that for any 2-primal ring R ,the polynomial ring R[ x] is not generalized n-clean.
In section three,semiclean rings will be extended to general rings.And semiclean general rings will be defined.A general ring I is called semiclean if (?)_x∈I,x = a+q,where a is periodic,i.e.,a~k = a~l,a∈I for some positive integers k and l ( k≠l) and q∈Q.In the other hand,some properties of semiclean rings will be dicussed.And it is proved that the matrix ring over semiclean ring is semiclean.
本文第二部分首先将n-clean环和广义clean环进行推广，引入一类新的环----广义n-clean环．如果(?)_x∈R，x =w+u_1+u_2+...+u_n，其中w是单位正则的，u_1,u_2,...,u_n∈U(R)．显然n-clean环和广义clean环一定是广义n-clean环，且举例说明广义n-clean环不一定是广义clean环．证明了幂等元是中心的，广义n-clean环是(n+1)-clean环．并讨论了广义n-clean的基本性质，以及与n-good环之间的关系．其次，主要讨论广义n-clean环的矩阵扩张．我们知道，任意环上的矩阵环M_m(R)是3-clean环，而文献[20]中证明了行列有限的矩阵环是2-clean环，因此，任意环R上的m×m矩阵环M_m(R)是2-clean的，本文采取对矩阵进行分解的方法，给出另外一种证明．最后，主要讨论了广义n-clean环的多项式扩张．证明环R是2-素环，多项式环R[x]不是广义n-clean环．
本文第三部分，主要是将半-clean环推广到一般环上，定义了半-clean一般环．称一般环I为半-clean的，如果(?)_x∈I，x=a+q，其中a是周期的，即对于某正整数k和l(k≠l)有a~k = a~l，a∈I，且q∈Q．最后论及研究了半-clean一般环的性质，并证明半-clean一般环上的矩阵环还是半-clean一般环．
In 1977,Nicholson proposed clean rings.A lot of algebra scholars at home and abroad begin to research clean rings.And as the generalization of unit regular rings,clean rings have practical application.Following clean rings,Xiao and Zhang generalized clean rings to n-clean rings and generalized clean rings.
In section two,firstly the notion of generalized n-clean rings is introduced.These rings are shown to be a natual generalization of n-clean rings and generalized clean rings.A ring R is called generalized n-clean if (?)_x∈R，x =w+u_1+u_2+...+u_n ,where w is a unit regular element and u_1,u_2,...,u_n∈U(R).Clearly n-clean rings and generalized clean rings must be generalized n-clean rings.And an example that generalized n-clean rings need not be generalized clean rings will be given.We will prove that if idempotents are central generalized n-clean rings are (n+1)-clean rings.The basic properties of generalized n-clean rings and the relationship between generalize n-clean rings and n-good rings will be discussed.Secondly,we will discuss matrix extensions of generalized n-clean rings.It is known that any matrix ring is 3-clean.And in [20],it is proved that the row and column-finite matrix ring is 2-clean.Thus any matrix ring is 2-clean.We will decompose the matrix and give another proof.Finally,we will mainly discuss polynomial extensions of generalized n-clean rings.It will be proved that for any 2-primal ring R ,the polynomial ring R[ x] is not generalized n-clean.
In section three,semiclean rings will be extended to general rings.And semiclean general rings will be defined.A general ring I is called semiclean if (?)_x∈I,x = a+q,where a is periodic,i.e.,a~k = a~l,a∈I for some positive integers k and l ( k≠l) and q∈Q.In the other hand,some properties of semiclean rings will be dicussed.And it is proved that the matrix ring over semiclean ring is semiclean.
引文
[1] Anderson F W, Fuller K R. Rings and categories of modules. Springer-Verlag, 1992.
[2] Goodearl K R. Von Neumann regular rings. Pitman, London, 1979.
[3] Tuganbaev A A. Rings close to regular. Kluwer Academic Publishers, 2002.
[4] Nicholson W K. Lifting idempotents and exchange rings. Trans.Amer.Math.Soc., 1977, 229: 269-278.
[5] Xiao G S, Tong W T. n-clean rings and weakly unit stable range rings. Comm.Algebra, 2005, 33(5): 1501-1517.
[6] Ehrlich G. Unit regular rings. Portugal Math., 1968, 27: 209-212.
[7] Henriksen M. Two classes of rings generated by their units. Journal of Algebra, 1974, 31: 182-193.
[8] Raphael R. Rings which are generalized by their units. Journal of Algebra, 1974, 28: 199-205.
[9] Handelman D. Perspectivity and cancellation in regular rings. Journal of Algebra, 1977, 48: 1-16.
[10] Vamos P. 2-good rings. Quart.J.Math., 2005, 56: 417-430.
[11] Fisher J W, Snider R L. Rings generalized by their units. Journal of Algebra, 1976, 42: 363-368.
[12] Gillman L, Jerison M. Rings of continuous functions. The University Series in Higher Maths, D.Van Nostrand Co., Princeton, N.J., 1960.
[13] Goodearl K R, Menal P. Stable range one for rings with many units. J.Pure Applied Algebra, 1988, 54: 261-287.
[14] Camillo V P, Yu H P. Exchange rings, units and idempotents. Comm.Algebra, 1994, 22: 4737-4749.
[15] Xiao G S, Tong W T. n-clean rings. Algebra Colloquium, 2006, 13: 599-606.
[16] Zhang H B, Tong W T. Generalized clean rings. J.of Nanjing University Math.Biquarterly, 2005, 22(2): 183-188.
[17] Ye Y Q. Semiclean rings. Comm.Algebra, 2003, 31(11): 5609-5625.
[18] Nicholson W K, Zhou Y. Clean general rings. Journal of Algebra, 2005, 291: 297-311.
[19]王静. n-clean一般环.甘肃科学学报, 2007, 19: 26-29.
[20] Wang Z, Chen J L. 2-clean Rings. Canad.Math.Bull.Vol., 2009, 52(1): 145-153.
[21]童晓平.单位正则环.浙江大学学报, 2000, 27(4): 377-381.
[22] Han J, Nicholson W K. Extensions of clean rings. Comm.Algebra, 2001, 29(6): 2589-2595.
[23] Chen W X. P-clean rings. International Journal of Mathematics and Mathematical Sciences, 2006,Article ID 75247, Pages 1-7.
[24] Kamal A A M. Some remarks on ore extension rings. Comm.Algebra, 1994, 22(10): 3637-3667.
[25]汪小琳,宋雪梅.关于σ-左半中心幂等元和斜多项式环的性质.西北师范大学学报, 2002, 38(2): 23-25.
[26] Marks G. Direct product and power series formations over 2-primal rings. in: S.K.Jain, S.T.Rizvi(Eds), Advances in Ring Theory, in: Trends in Math, Birkhauser, Boston, 1997, 239-245.
[27] Marks G. A taxonomy of 2-primal rings. Journal of Algebra, 2003, 266: 494-520.
[28] Chen W. A note on polynomial rings. to appear in Southeast Asian Bulletin of Mathematics.
[29] Anderson D D, Camillo V P. Commutative rings whose elements are a sum of a unit and idempotent. Comm.Algebra, 2002, 30(7): 3327-3336.
[30] Nicholson W K, Zhou Y. Rings in which elements are uniquely the sum of an idempotent and a unit. Glasgow Math.J., 2004, 46: 227-236.
[31] Marks G. Skew polynomial rings over 2-primal rings. Comm.Algebra, 1999, 27(9): 4411-4423.
[32] Liu Z K. Special properties of rings of generalized power series. Comm.Algebra, 2004, 32(8): 3215-3226.
[33] Liu Z K. The ascending chain condition for principal ideals of rings of generalized power Series. Comm.Algebra, 2004, 32(9): 3305-3314.
[34] Jerry D, Mary P. Ideals and quotient rings of skew polynomial rings. Comm.Algebra, 1993, 21(3): 799-823.
[35]姜侠.一般clean环的扩张.兰州交通大学学报, 2006, 25(3): 149-150.
[36] Camillo V, Khurana D A. A characterization of unit regular rings. Comm.Algebra, 2001, 29: 2293-2295.
[37] Ara P. Extensions of exchange rings. Journal of Algebra, 1997, 197: 409-423.
[38] Ratinov. The structure of semiperfect rings with commutative Jacobson radical. Mathematics of the USSR-Sbornik, 1981, 38(3): 427-436.
[39] Chen H. Exchange rings with artinian primitive factors. Algebra Represent Theory, 1999, 2: 201-207.
[40] Warfield R B. Exchange rings and decompositions of modules. Math.Ann., 1972, 199: 31-36.
[41] Nicholson W K. Strongly clean rings and Fitting’s lemma. Comm.Algebra, 1999, 27(8): 3583-3592.
[42] Ara P, Goodearl K R O’Meara K C, Pardo E. Separative cancellation for projective modules overexchange rings. Israel J.Math., 1998, 105: 105-137.
[43] Wang Z, Chen J L. On strongly clean general rings. Journal of Mathematical Research and Exposition, 2007, 27(1): 28-34.
[44] Wang Z, Chen J L. Extensions of stronglyπ-regular general rings. Jounal of Southeast University, 2007, 23(2): 309-312.
[45]崔书英,陈卫星. clean一般环的几个结果.大学数学, 2008, 24(1): 66-69.
[46] Evans E G. Krull-Schmidt and cancellation over local rings. Pacifica J.Math, 1973, 46: 115-121.
[47] Ara P. Stronglyπ-regular rings have stable range one. Proc.A.M.S., 1996, 124: 3293-3298.
[48] Chen H. On stable range conditions. Comm.Algebra, 2000, 28(12): 3913-3924.
[2] Goodearl K R. Von Neumann regular rings. Pitman, London, 1979.
[3] Tuganbaev A A. Rings close to regular. Kluwer Academic Publishers, 2002.
[4] Nicholson W K. Lifting idempotents and exchange rings. Trans.Amer.Math.Soc., 1977, 229: 269-278.
[5] Xiao G S, Tong W T. n-clean rings and weakly unit stable range rings. Comm.Algebra, 2005, 33(5): 1501-1517.
[6] Ehrlich G. Unit regular rings. Portugal Math., 1968, 27: 209-212.
[7] Henriksen M. Two classes of rings generated by their units. Journal of Algebra, 1974, 31: 182-193.
[8] Raphael R. Rings which are generalized by their units. Journal of Algebra, 1974, 28: 199-205.
[9] Handelman D. Perspectivity and cancellation in regular rings. Journal of Algebra, 1977, 48: 1-16.
[10] Vamos P. 2-good rings. Quart.J.Math., 2005, 56: 417-430.
[11] Fisher J W, Snider R L. Rings generalized by their units. Journal of Algebra, 1976, 42: 363-368.
[12] Gillman L, Jerison M. Rings of continuous functions. The University Series in Higher Maths, D.Van Nostrand Co., Princeton, N.J., 1960.
[13] Goodearl K R, Menal P. Stable range one for rings with many units. J.Pure Applied Algebra, 1988, 54: 261-287.
[14] Camillo V P, Yu H P. Exchange rings, units and idempotents. Comm.Algebra, 1994, 22: 4737-4749.
[15] Xiao G S, Tong W T. n-clean rings. Algebra Colloquium, 2006, 13: 599-606.
[16] Zhang H B, Tong W T. Generalized clean rings. J.of Nanjing University Math.Biquarterly, 2005, 22(2): 183-188.
[17] Ye Y Q. Semiclean rings. Comm.Algebra, 2003, 31(11): 5609-5625.
[18] Nicholson W K, Zhou Y. Clean general rings. Journal of Algebra, 2005, 291: 297-311.
[19]王静. n-clean一般环.甘肃科学学报, 2007, 19: 26-29.
[20] Wang Z, Chen J L. 2-clean Rings. Canad.Math.Bull.Vol., 2009, 52(1): 145-153.
[21]童晓平.单位正则环.浙江大学学报, 2000, 27(4): 377-381.
[22] Han J, Nicholson W K. Extensions of clean rings. Comm.Algebra, 2001, 29(6): 2589-2595.
[23] Chen W X. P-clean rings. International Journal of Mathematics and Mathematical Sciences, 2006,Article ID 75247, Pages 1-7.
[24] Kamal A A M. Some remarks on ore extension rings. Comm.Algebra, 1994, 22(10): 3637-3667.
[25]汪小琳,宋雪梅.关于σ-左半中心幂等元和斜多项式环的性质.西北师范大学学报, 2002, 38(2): 23-25.
[26] Marks G. Direct product and power series formations over 2-primal rings. in: S.K.Jain, S.T.Rizvi(Eds), Advances in Ring Theory, in: Trends in Math, Birkhauser, Boston, 1997, 239-245.
[27] Marks G. A taxonomy of 2-primal rings. Journal of Algebra, 2003, 266: 494-520.
[28] Chen W. A note on polynomial rings. to appear in Southeast Asian Bulletin of Mathematics.
[29] Anderson D D, Camillo V P. Commutative rings whose elements are a sum of a unit and idempotent. Comm.Algebra, 2002, 30(7): 3327-3336.
[30] Nicholson W K, Zhou Y. Rings in which elements are uniquely the sum of an idempotent and a unit. Glasgow Math.J., 2004, 46: 227-236.
[31] Marks G. Skew polynomial rings over 2-primal rings. Comm.Algebra, 1999, 27(9): 4411-4423.
[32] Liu Z K. Special properties of rings of generalized power series. Comm.Algebra, 2004, 32(8): 3215-3226.
[33] Liu Z K. The ascending chain condition for principal ideals of rings of generalized power Series. Comm.Algebra, 2004, 32(9): 3305-3314.
[34] Jerry D, Mary P. Ideals and quotient rings of skew polynomial rings. Comm.Algebra, 1993, 21(3): 799-823.
[35]姜侠.一般clean环的扩张.兰州交通大学学报, 2006, 25(3): 149-150.
[36] Camillo V, Khurana D A. A characterization of unit regular rings. Comm.Algebra, 2001, 29: 2293-2295.
[37] Ara P. Extensions of exchange rings. Journal of Algebra, 1997, 197: 409-423.
[38] Ratinov. The structure of semiperfect rings with commutative Jacobson radical. Mathematics of the USSR-Sbornik, 1981, 38(3): 427-436.
[39] Chen H. Exchange rings with artinian primitive factors. Algebra Represent Theory, 1999, 2: 201-207.
[40] Warfield R B. Exchange rings and decompositions of modules. Math.Ann., 1972, 199: 31-36.
[41] Nicholson W K. Strongly clean rings and Fitting’s lemma. Comm.Algebra, 1999, 27(8): 3583-3592.
[42] Ara P, Goodearl K R O’Meara K C, Pardo E. Separative cancellation for projective modules overexchange rings. Israel J.Math., 1998, 105: 105-137.
[43] Wang Z, Chen J L. On strongly clean general rings. Journal of Mathematical Research and Exposition, 2007, 27(1): 28-34.
[44] Wang Z, Chen J L. Extensions of stronglyπ-regular general rings. Jounal of Southeast University, 2007, 23(2): 309-312.
[45]崔书英,陈卫星. clean一般环的几个结果.大学数学, 2008, 24(1): 66-69.
[46] Evans E G. Krull-Schmidt and cancellation over local rings. Pacifica J.Math, 1973, 46: 115-121.
[47] Ara P. Stronglyπ-regular rings have stable range one. Proc.A.M.S., 1996, 124: 3293-3298.
[48] Chen H. On stable range conditions. Comm.Algebra, 2000, 28(12): 3913-3924.