一些环的交换性条件
|
摘要
本文通过对Kothe半单纯环、半质环乃至任意环的研究,得到了一些环的交换性条件.其主要结果如下:
首先我们证明了任何满足两项多项式为零的无零因子环是交换的.并利用此得到:若为半质环，且对所有有则为交换环.其中满足下列条件之一：
1)与开始文字与结尾文字不完全相同，且首项系数均为1；
2)含(或)的因子，既不含因子也不含因子；
3)中含(或)因子， 不含(或)因子
则为交换环.
此结论推广了包括Quadri、邱琦章及郭元春等人得出的十几个结果.
其次,我们证明了:若为K?the半单纯环,且对,有依于的非负整数及使得
对一切的成立,则为交换环.
再次我们得到满足条件(A)的K?the半单纯环是交换的.(其中条件(A)是指: 若对任意的，存在依赖于的整系数多项式，形如，为一整数多项式，其每一项关于的次数不小
于，关于的次数不小于（此处为依赖于的正整数），，使.)
最后我们证明了:若是对具有强性质的关于一个文字的齐次多项式,,且对,存在自然数,使得
则当中不含对次数小于k-1的项,且时,R为交换环；当k=1或R有1时,只需对具有强性质即可.
We get some new results on commutativity of rings by studying the K?the semisimple ring, semiprime ring and arbitrary ring. The main results in this paper is as following:
First we prove that if a ring has no zero-divisors and satisfies a center-polynomial of two terms, then the ring is commutative. In view of it we get that a semiprime ring is commutative if it satisfy for any . must satisfy one of the following conditions:
1) The beginning indeterminate and the terminal indeterminate of are not as same as that of , and the all coefficients of the first term in and are 1;
2) contains the factor of or , contains neither nor ;
3) contains the factor of (or ), contains neither nor .
Second we have that if R is a K?the semisimple ring and for any , there exists nonnegative integers and polynomial depending on ,such that
,
then R is commutative.
Third we have if R is a K?the semisimple ring and satisfies condition (A), then the ring is commutative. (The condition (A) means: for any there exists a polynomial depending on , such that . Where and is a integer coefficient polynomial, in which the degree of the indeterminate is no less then 2 and the degree of the indeterminate is no less then . and is a nonnegative integer depending on and ).
Last we prove that if is a polynomial with integer coefficient that satisfies strong -condition on the set and homogeneous about one indeterminate and for any , there exists an integer , such that , where and degree of in is no less then k-1. Then R is commutative.
首先我们证明了任何满足两项多项式为零的无零因子环是交换的.并利用此得到:若为半质环，且对所有有则为交换环.其中满足下列条件之一：
1)与开始文字与结尾文字不完全相同，且首项系数均为1；
2)含(或)的因子，既不含因子也不含因子；
3)中含(或)因子， 不含(或)因子
则为交换环.
此结论推广了包括Quadri、邱琦章及郭元春等人得出的十几个结果.
其次,我们证明了:若为K?the半单纯环,且对,有依于的非负整数及使得
对一切的成立,则为交换环.
再次我们得到满足条件(A)的K?the半单纯环是交换的.(其中条件(A)是指: 若对任意的，存在依赖于的整系数多项式，形如，为一整数多项式，其每一项关于的次数不小
于，关于的次数不小于（此处为依赖于的正整数），，使.)
最后我们证明了:若是对具有强性质的关于一个文字的齐次多项式,,且对,存在自然数,使得
则当中不含对次数小于k-1的项,且时,R为交换环；当k=1或R有1时,只需对具有强性质即可.
We get some new results on commutativity of rings by studying the K?the semisimple ring, semiprime ring and arbitrary ring. The main results in this paper is as following:
First we prove that if a ring has no zero-divisors and satisfies a center-polynomial of two terms, then the ring is commutative. In view of it we get that a semiprime ring is commutative if it satisfy for any . must satisfy one of the following conditions:
1) The beginning indeterminate and the terminal indeterminate of are not as same as that of , and the all coefficients of the first term in and are 1;
2) contains the factor of or , contains neither nor ;
3) contains the factor of (or ), contains neither nor .
Second we have that if R is a K?the semisimple ring and for any , there exists nonnegative integers and polynomial depending on ,such that
,
then R is commutative.
Third we have if R is a K?the semisimple ring and satisfies condition (A), then the ring is commutative. (The condition (A) means: for any there exists a polynomial depending on , such that . Where and is a integer coefficient polynomial, in which the degree of the indeterminate is no less then 2 and the degree of the indeterminate is no less then . and is a nonnegative integer depending on and ).
Last we prove that if is a polynomial with integer coefficient that satisfies strong -condition on the set and homogeneous about one indeterminate and for any , there exists an integer , such that , where and degree of in is no less then k-1. Then R is commutative.
引文
1 Peric,veselin.Some commutativity results for s-unital rings with constraints on four elements subsets.Math. 1999,(10):43-57.
2 Abujabal,Hamza A.S.Commutativity of rings via Streb's classification Southeast Asian Bull .Math,1998,(1): 1-10.
3 Ashraf.Mohammad.On commutativity of s-unital rings with commutator constraints.Far East Math.Sci.1998,Special Volume partⅡ:163-172
4 Kezlan,Thomasp.On additive epimorphisms and commutativety of rings.Ⅱ.Proc,Roy.Irish Acad,Sect, 1996,(2):139-146.
5 Psomopoulos,Evagelos.commutativity of torsion-free periodic rings. Math.Montisnigri,1998,(9):87-91.
6 Abujabal,H.A.S Ashraf,Mohammad.On commutativity of rings involving certain polynomial constraints.AlgeBra Colloq, 1998,(1): 111-116.
7 Ashraf,M.Some commutativity theorems for associative rings Rend. Sem. Mat.Univ.Politec.Torino,1997,(2):99-108.
8 Komatsu,Hiroaki.Commutativity of rings with powers commuting on subsets.Math.Okayama Uniw,1997,(39):45-56.
9 Bell,Howard.E.Lucier,Jason.On additive Maps and commutativity in rings,Results Math,1999,(2):1-8.
10 Cohn.P.M.Reversible rings.Bull London Math.Soc., 1999,(6):641-648.
11 Quadri, Murtaza A, Achlesh. A note on co-commutativiy preserving derivations.Aligarh,Bull.Math.,1995/96,(16):71-74.
12 Awtar.R.A remark on the commutativity of certain rings. Proc Amer Math, 1973, (41): 370-372.
13 Chen HuanYin.Exchange rings related comparability and power
substitution. Communications in Algebra, 1998, (10): 3383-3410.
14 Zhang Shouchuan.The Bare and Jacobson radicals of crossed products. Acta Mathematica Hungarica, 1998, (78): 11-12.
15 孟宪利. 半质环的若干交换性条件.工科数学,1977,(3):37-39.
16傅昶林,王亚芳,田淑荣.中心多项式与环的交换性.数学研究与评论,2001,(4):607-610.
17 邓清.素环的对称双导与交换性.数学研究与评论,1996,(3):427-430.
18 朱金涛.半质环与极大右零化子.武汉工学院学报,1995,(3):90-94.
19 FU Changlin.Commutative Conditions for Rings with [J].Northeastern Math,1991，7(2)：206-208.
20杨新松,张海燕.零因子分布对环交换性的影响.哈尔滨理工大学学报,1999,(3):17-19.
21 郭华光.半质环的几个交换性定理.数学研究与评论,1997,(3):423-426.
22 戴跃进.结合环的几个交换性条件.福建师范大学学报,1996,(1):30-32.
23 田淑荣.半质环的交换性.哈尔滨理工大学学报,1998,5:100-102.
24于宪君,朱杰.结合环的交换性条件.黑龙江大学自然科学学报,1999,(2): 22-28.
25 蔡敏,傅昶林.半质环的交换性条件.数学杂志,2000,(4):417-420.
26于宪君,朱杰.结合环的几个交换性条件.哈尔滨师范大学自然科学学报,2000,(3):8-12.
27 谢邦杰.抽象代数.第二版.上海科技出版社,1982.
28 Herstein I.N.Tow remarks on the commutativity of rings. Canad. J.Math,1955, (7): 411-412.
29 郭元春.结合环的几个交换性定理.吉林大学自然科学学报，1982，
(3):13-16.
30 Quadri M.A.A note on commutativity of semiprime rings. Math, 1978, (22): 509-510.
31 朱孝璋.Bear半单纯环的几个交换性条件.吉林大学自然科学学报,1984,(3):27-34.
32 邱琦章.结合环的若干交换性条件.数学杂志,1984,(4):31-39.
33 邱琦章.半单纯环交换性的充分条件.数学杂志,1982,(3):291-301.
34 Quadri M.A.Some polynomial identities and commutativity of semiprime rings. Tamkang J.Math.1987, 18(4): 19-22.
35 于宪君,朱杰.结合环的交换性条件.黑龙江大学自然科学学报,1999,(6):22-28.
36王湘浩.关于K?the半单纯环.东北人民大学学报(自然科学版),1995,(1):143-147.
37 戴跃进.某些环的交换性条件.数学杂志,1994,(3):431-434.
38 戴跃进.K?the半单纯环的几个交换性定理.数学杂志, 1991,(3):331-334.
39 刘则毅.结合环的几个交换性条件.吉林大学自然科学学报,1986,(1): 44-54.
40 傅昶林、郭元春.半质环的交换性条件.数学学报,1995,(2):242-247.
41 熊全淹.环构造.第二版.湖北教育出版社.1985.
2 Abujabal,Hamza A.S.Commutativity of rings via Streb's classification Southeast Asian Bull .Math,1998,(1): 1-10.
3 Ashraf.Mohammad.On commutativity of s-unital rings with commutator constraints.Far East Math.Sci.1998,Special Volume partⅡ:163-172
4 Kezlan,Thomasp.On additive epimorphisms and commutativety of rings.Ⅱ.Proc,Roy.Irish Acad,Sect, 1996,(2):139-146.
5 Psomopoulos,Evagelos.commutativity of torsion-free periodic rings. Math.Montisnigri,1998,(9):87-91.
6 Abujabal,H.A.S Ashraf,Mohammad.On commutativity of rings involving certain polynomial constraints.AlgeBra Colloq, 1998,(1): 111-116.
7 Ashraf,M.Some commutativity theorems for associative rings Rend. Sem. Mat.Univ.Politec.Torino,1997,(2):99-108.
8 Komatsu,Hiroaki.Commutativity of rings with powers commuting on subsets.Math.Okayama Uniw,1997,(39):45-56.
9 Bell,Howard.E.Lucier,Jason.On additive Maps and commutativity in rings,Results Math,1999,(2):1-8.
10 Cohn.P.M.Reversible rings.Bull London Math.Soc., 1999,(6):641-648.
11 Quadri, Murtaza A, Achlesh. A note on co-commutativiy preserving derivations.Aligarh,Bull.Math.,1995/96,(16):71-74.
12 Awtar.R.A remark on the commutativity of certain rings. Proc Amer Math, 1973, (41): 370-372.
13 Chen HuanYin.Exchange rings related comparability and power
substitution. Communications in Algebra, 1998, (10): 3383-3410.
14 Zhang Shouchuan.The Bare and Jacobson radicals of crossed products. Acta Mathematica Hungarica, 1998, (78): 11-12.
15 孟宪利. 半质环的若干交换性条件.工科数学,1977,(3):37-39.
16傅昶林,王亚芳,田淑荣.中心多项式与环的交换性.数学研究与评论,2001,(4):607-610.
17 邓清.素环的对称双导与交换性.数学研究与评论,1996,(3):427-430.
18 朱金涛.半质环与极大右零化子.武汉工学院学报,1995,(3):90-94.
19 FU Changlin.Commutative Conditions for Rings with [J].Northeastern Math,1991，7(2)：206-208.
20杨新松,张海燕.零因子分布对环交换性的影响.哈尔滨理工大学学报,1999,(3):17-19.
21 郭华光.半质环的几个交换性定理.数学研究与评论,1997,(3):423-426.
22 戴跃进.结合环的几个交换性条件.福建师范大学学报,1996,(1):30-32.
23 田淑荣.半质环的交换性.哈尔滨理工大学学报,1998,5:100-102.
24于宪君,朱杰.结合环的交换性条件.黑龙江大学自然科学学报,1999,(2): 22-28.
25 蔡敏,傅昶林.半质环的交换性条件.数学杂志,2000,(4):417-420.
26于宪君,朱杰.结合环的几个交换性条件.哈尔滨师范大学自然科学学报,2000,(3):8-12.
27 谢邦杰.抽象代数.第二版.上海科技出版社,1982.
28 Herstein I.N.Tow remarks on the commutativity of rings. Canad. J.Math,1955, (7): 411-412.
29 郭元春.结合环的几个交换性定理.吉林大学自然科学学报，1982，
(3):13-16.
30 Quadri M.A.A note on commutativity of semiprime rings. Math, 1978, (22): 509-510.
31 朱孝璋.Bear半单纯环的几个交换性条件.吉林大学自然科学学报,1984,(3):27-34.
32 邱琦章.结合环的若干交换性条件.数学杂志,1984,(4):31-39.
33 邱琦章.半单纯环交换性的充分条件.数学杂志,1982,(3):291-301.
34 Quadri M.A.Some polynomial identities and commutativity of semiprime rings. Tamkang J.Math.1987, 18(4): 19-22.
35 于宪君,朱杰.结合环的交换性条件.黑龙江大学自然科学学报,1999,(6):22-28.
36王湘浩.关于K?the半单纯环.东北人民大学学报(自然科学版),1995,(1):143-147.
37 戴跃进.某些环的交换性条件.数学杂志,1994,(3):431-434.
38 戴跃进.K?the半单纯环的几个交换性定理.数学杂志, 1991,(3):331-334.
39 刘则毅.结合环的几个交换性条件.吉林大学自然科学学报,1986,(1): 44-54.
40 傅昶林、郭元春.半质环的交换性条件.数学学报,1995,(2):242-247.
41 熊全淹.环构造.第二版.湖北教育出版社.1985.