素环的对称双导与交换性

Symmetric Bi-derivations and Commutativity of Prime Rings

中文关键词:  素环，交换性，对称双导．

英文关键词:prime ring, commutativity, symmetric bi-derivation.

中文摘要:

      本文讨论当一个素环容许一个非零对称双导时，其迹函数与此环的交换性的关系，得到了与［４，５］类似的结果．

英文摘要:

      Let R be a ring with center Z(R) . A mapping D:R×R→R is called a symmetric bi-derivation, if D(x,y)=D(y,x),D(x+y,z)=D(x,z)+D(y,z) and D(xy , z ) = D (x , z ) y+ xD (y , z ) for all x , y , z ∈R . We show that a prime ring R with char R ≠ 2, 3, admitting a nonzero symmetric bi-derivation D , is commutative if either [ x²,D (x , x ) ] ∈ Z (R ) for all x∈R or x D (x , x ) ±D (x , x ) x ∈ Z (R ) for all x ∈R.