On The Ring Of Morita Context

Morita Context was introduced initially by Morita in order to prove WedderburnTheorem about the structure of simple ring. Later, it became basic tool as various formsfor reseaching primitive ring with minimal one-side ideal. Morita Context is an importanttool, by which we can get many useful results: Goldie's Theorem about semi-prime ideal;Webberburn's Structure Theorem of semi-simple artinian ring; make use of Morita Contextring's propositions to study proposition relations between A#H and A~H (when H is Hopfalgebra or weak Hopf algebra respectively)……. Also, people often make use of MoritaContext ring to construct examples of rings which with some propositions in ring theory.In recent years, there are many papers making research into Morita Context from kindsof different directions [31-33]. In this paper, our work is mainly around Morita Context.Section one mainly introduce relative definition, proposition and applications aboutMorita Context ring.Section two make a study of 3-order Morita Context ring. As we know, Morita Con-text ring established relations between two rings which with weaker conditions than thatof two Morita equivalent rings. Two Morita equivalent rings have many good propositions.In fact, we found that though the former did not have propositions as perfect as that of thelatter, the former still have some good propositions. Draw inspiration from this, we enterthe definition of 3-order Morita Context, namely, extending the object of study from tworings to three rings. In addition, we gain enlightenment from the sufficient and necessarycondition on when the two rings in Morita Context ring became Morita equivalent rings,we try start from 3-order Morita Context ring to definite the 3-pair Morita equivalentamong three rings.That is to say, constructed the definition of triple Morita equivalent.Then we pay attention to the proposition relation among 3-order Morita Context ring Cand rings: R, S, T. We mainly make a study of propositionrelations when the ring withfollowing conditions respectively:Reversible ring,ZC_nring,Abelian ring,quasi-Duo ring,left V-ring,P.P.-ring,primitivering,semiregular ring,Suitable ring,semi-local ring,(semi)perfect ring,semiprimaryring,generalized Hopf ring,Clean ring,PS-ring,Rigid ring,reduced ring,local reducedring,direct finite ring,Noether ring,ring satisfing m-fold stable range condition,ringsatisfing unit 1-stable range condition and ring satisfing n-stable range condition.