Abstract: A quasi-dual bimodule SMR can be characterized as every essential submodule K of MR and every essential left ideal L of S which satisfy rMlS(K)=K and lSrM(L)=L respectively.The relations are discussed between quasi-dual 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 MR 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 MR,lS(A∩B)=lS(A)+lS(B).

Key words: quasi-dual bimodules, dual-bimodules, annihilators, M-minimal-injective modules