导子与环的交换性

GOMMUTATIVITY OF RINGS WITH DERIVATIONS

讨论了带有非零导子的结合环的交换性,证明了:定理1 R是特征非2的素环,f,g为R的两个非零导子,若有自然数n使得x^nfg(y)-fg(y)x^n∈Z(R) (?)x,y∈R则R可换.定理3 R为无零因子环,d为R的非零导子,若(?)x∈R,d^n_x∈Z(R)且R的特征不是(n+1)1的因子,则R可换.定理5 若素环R的特征不为2,U为R的非零Lie理想,且(?)u∈U有udu+duu∈Z(R),则u^2∈Z(R)且当u^2∈U时,U(?)Z(R).

In this paper, the commutativity of rings with nonzero derivations is discussed. The main theorems are the following:Theorem 1 Let R be a prime ring with CharR≠2. and f,g be nonzero derivations of R, If there is a fixed natural number n, such thatfor all x and y in R, then R is commutative.Theorems Let R be a ring without zerodivisors, n be a fixed natural number. If dnx∈Z(R) for all x in R,and Char R is not a divisor of (n+1)1. then R is commutative.Theorems Let R be a prime ring with Char R≠2, U be a nonzero Lie ideal of R, and d be a nonzero derivation. If udu+du.u∈Z(R) for all u in U.then u2∈Z(R) and when u2∈U,U Z(R).