拟对偶双边模和对偶双边模

拟对偶双边模SMR可以被刻画成MR的每一个本质子模K和S的所有本质左理想L分别满足rMlS(K)=K和lSrM(L)=L.拟对偶双边模和对偶双边模的关系表明：一个左拟对偶双边模SMR如果满足下列条件之一，则它成为坐对偶双边模：(1)SM 是单内射的并且MR是一个M-单内射kasch-模； （2）MR是一个M-单内射kasch-模并且对SS 的任意2个理想L1和L2 有rM(L1∩L2)=rM(L1)+rM(L2)；（3）SM是单内射的并且对MR的任意2个子模A和B，有lS(A∩B)=lS(A)+lS(B).

Quasi-Dual Bimodules and Dual - Bimodules

A quasi-dual bimodule sMR can be characterized as every essential submodule K of MR and everyessential left ideal L of S which satisfy rMls (K)=K and lsrM(L) = L respectively. The relations are discussed between quasidual bimodules and dual-bimodules;a left quasi-dual bimodule is a left dual bimodule if it satisfies one of the following conditions:(i) sM is minimal injective and Ma is a M- minimal injective kasch- module; (ii) MR is a M-minimal injective kasch - module and rM ( L1∩L2 )= rM (L1)+rM (L2) for any two ideals L1 and L2 of sS;(iii) sM is minimal injective and for any two submodules A and B of Ma,Is(A∩B) =Is(A) +Is(B).