一类特殊近环的序和导子

ORDER AND DERIVATION IN A SPECIAL CLASS OF NEAR-RINGS

我们引入一类非结合近环－零积结合分配生成近环，研究它的Ａｂｉａｎ序关系和导子．我们的主要结果是：（１）设Ｘ是零积结合分配生成约化近环Ｎ的子集，ｃ∈Ｎ，则ｃ＝ＳｕｐＸ当且仅当ｃ是Ｘ的一个上界且Ａ（Ｘ）＝Ａ（ｃ）；（２）设Ｘ＝｛ｘｉ｜ｉ∈Ｉ｝，Ｙ＝｛ｙｉ｜ｊ∈Ｊ｝是Ｎ的两个正交子集，ＳｕｐＸ＝ｘ，ＳｕｐＹ＝ｙ，Ｚ＝｛ｘｉｊｉ｜ｉ∈Ｉ，ｊ∈Ｊ｝，则Ｚ是Ｎ的一个正交子集且ＳｕｐＺ＝ｘｙ；（３）一个挠自由零积结合分配生成约化近环不容纳一个非零的幂零导子。

in this paper we introduce a class of nonassociative near -rings,zero-product-associative distributively generated near-ring,and study Abian's order relation and derivation in it.Our main results are:(1)let X be a subset of N which is zero-product-associative distributively generated reduced near-ring,and ∈N,then c=Sup X if and only if c is an upper of X and A(X)=A(c);(2) let X={xi|i∈I} and Y={yj|j∈J} be two orthogonal subsets of N,Sup X =x,Sup Y=y and Z= (xiyj|i∈I,j∈J},then Z is an orthogonal subset of N and Sup Z=xy;(3)a torsion-free,zero-product-associative distributively generated reduced near-ring admits no a nonzero nilpotent derivation.