结合环的几个交换性条件

Some Commutaticity Conditions on Associative Rings

设Ｚ（Ｒ）为环Ｒ的中心．本文证明了满足下列条件之一的环Ｒ必为交换环：（１）Ｒ是Ｋｏｅｔｈｅ半单纯环，且对任意ａ，ｂ∈Ｒ，均存在一个非负整数Ｋ＝Ｋ（ａ，ｂ）及一个整系数多项式ｆｘ（ｘ．ｙ）（它的每一个单项式均含有ｓ＝ｓ（ａ，ｂ）（＞１）个ｘ和ｔ＝ｔ（，ｂ）（≥Ｋ）个ｙ）使得ａｈＫ－ｆｘ（ａ，ｂ）∈Ｚ（Ｒ）；（２）Ｒ是Ｂａｅｒ半单纯环，且对任意ａ，ｂ∈Ｒ．均存在一个非负整数Ｋ＝Ｋ（ａ，ｂ）≤ｎ及一整系数多项式ｆｘ（ｘ，ｙ）（它的每一个单项式均含有ｓ＝ｓ（ａ，ｂ）（＞１）个ｘ和ｔ＝ｔ（ａ，ｂ）（≥Ｋ）个ｙ），使得ａｂＫ－ｆｘ（ａ．ｂ）∈Ｚ（Ｒ）．其中ｎ是一个固定的正整数，ｆｘ（ｘ，ｙ）可随Ｘ＝（ａ，ｂ）而变．

in this paper,one proveds that a ring R with centre Z(R) is commutative if it satisfies one of the following conditions: (1 )R is a Koethe semisimple ring,and for arbitrary a,b in R, there exist a non-negative integer K=K(a, b) and a polynomial fx(x,y) (every monomial of which contains s=s(a,b) (>1 )x's and t=t(a,b) (≥K ) y's)such that abK --f x (a,b) ∈ Z(R),fx (x,y) can vary as x= (a,b) varies. (2)R is a Baer semisimple ring,and for arbitrary a,b in R,there exist a non--negative integer K (a,b) (≤n ) and a polynomialfx (x,y) (every monomial of which contains s=s(a,b) (>1 )X's and t=t(a,b) (≥K)y's)such that abK --f x (a,b) ∈ Z (R), where n is a fixed positive integer,fx (x .y) can vary as x= (a,b) varies.