摘要
设R是一个结合环,满足由2x=0,x∈R,可推出x=0,N是R的一个非零理,D_1,D_2是R的二个约当微商,使D_1(N)和D_2(N)分别含有R的一个交换子正则元,且对任意a,b∈N,都有D_1(a)D_2(b)=D_2(b)D_1(a),则R是交换环。
In this paper the following results are proved:
1. Let R be an associative ring in which 2x=0 implies x=0, and N be
an (nonzero) ideal of R and D_1, D_2 be two Jordan derivations on R, such that
D_1(N) and D_2(N) contain a regular element of R respectively. If R satisfies
D_1 (α)D_2 (b)=D_2 (b)D_1 (α)
for all α, b∈N, then R is commutative.
2. Let R be an associative ring and D be a nonzero mapping of R such
that
D (α+b) =D (α)+ D (b) and D (αb) = D (b)α+ bD (α)
If any T ideal(≠0) of R contains a regular element in R, the R is commu-
tative.
