Study On The Homological Classification Problems Of Condition(SP)

In this paper,we study the properties of condition(SP),and investigate the homological classification problem of condition(SP)on arbitrary acts,cyclic S-acts and Rees quotient acts.This paper is divided into two chapters.The first chapter defines the condition(SP),studies the implication relation among the condition(SP),the strong flatness,the condition(P)and the condition(E),and gives corresponding examples to illustrate them.Second chapter considers the homological classification problem of condition(SP).In the first section,we introduce the definition of cyclic acts,flatness,weakly flatness,principal weakly flatness,torsion free,right PP monoid,and chain elements.In the second section,using the relation between condition(SP)and other flat properties,we study the homological classification problem of condition(SP)in arbitrary acts.In the third section,we get the characterization of monoids on torsion free,principal weakly flatness,weakly flatness and flatness satisfying the condition(SP).In the fourth section,we give the homological clasification results of all Rees quotient acts on condition(SP).