关于环R[D,C]的伪morphic性

Pseudo-Morphic Property of R[D,C] Rings

R称为左伪morphic环,若对任意的a∈R,存在b,c∈R使得Ra=l(b),Rb=l(c),其中l(b),l(c)表示R中元素b且c的左零化子.本文主要研究R[D,C]环的伪morphic性,证明了环R[D,C]是左伪morphic的当仅当(1)D是左伪morphic环;(2)对任意的x∈C,存在y∈C使得Cx=lC(y),Dx=lD(y).受文[2]的启发,定义了左[D,C]-伪morphic元,并研究了这类元素的性质.

A ring R is called left pseudo-morphic if for everya∈ R,there exist b,c∈ R such that Ra =l（b）,Rb =l（c）,wherel（b）,l（c）denote the left annihilator of b,c in R.In this paper,we characterize the pseudo-morphic properties of R[ D,C] rings.It is shown that R[ D,C] is aleft pseudo-morphic ring if and only if （1）D is a left pseudo-morphic ring;（2）for any x∈ C,there exissty∈ C such that Cx =lC（y） and Dx =lD（y）.By idea from [2],we define left R[ D,C]-pseudo morphic element.Further more,we discuss the properties of these elements.