高阶差分的表达式与环的交换条件

Expressions of Difference of Higher Order and a Condition for Commutativity of Rings

本文给出了非结合环上高阶差分的精确表达式,井由此证明了:设 R 为有1之非结合环,则当具有下列两条件时:(1)R 中任何元素 x,y 满足 F(x,y)=0,(2)R 是 P-扭自由的。R 是交换的。此处 P 是诸 P_i 的最大正公因数,而 P_i=(m_i-1)!(n_i-1)!q_i 适于1≤i≤d或 i=α,i=β。

in this paper exact expressions of difference of higher order on non-asso- ciative rings are given and the following theorem is proved. Theorem.Let R be a non-associative ring with 1.If R satisfies the follo- wing two conditons: (1)F(X,Y)=0 for all X,Y∈R. (2)R is a p-torsion free,where P is a positive maximal common factor in all p_i for p_i=(mi-i)!(n_i-1)!q_(?)and 1≤i≤d or i=α and i=β. Then R is commutative. Moreover the cases in Which R is an associative ring and semi-prime ring are respectively discussed.