The Balanced Pair And Hovey Triples Over Triangular Matrix Ring

(?)be a triangular matrix ring,where A and B are rings and U is a left B right A-bimodule.This paper we use balanced pairs,Hovey triples and Kaplansky class over A and B to study balanced pairs,Hovey triples and Kaplansky class over T.Let xA and yA be two classes of A-modules and xB and yB be two classes of B-modules.Firstly,we prove that under certain conditions,(xA,yA)and(xB,yB)respectively are balanced pairs over A and B if and only if(?)is balanced pair over T.And we prove that if(?)is a balanced pair over T,then(xA,yA)and(xB,yB)respectively are balanced pairs over A and B.Secondly,we study Hovey triple over triangular matrix ring,and use Hovey triple(CA,WA,FA)over A and Hovey triple(CB,WB,FB)over B to discuss the property of(?)over T,prove that this triple is a Hovey triple under certain conditions.Lastly,we investigate Kaplansky property of(?)and give judgment method that three classes(?)are Kaplansky classes.