The faithful bi-module ulcornerMulcorner with its endomorphism ring Endulcorner(M) such that Mulcorner is flat (in other words, Endulcorner(M)-flat, or endo-flat)and with a commutative ring R containing an identity has been studied in this paper. The chain conditions of a faithful endo-flat module ulcornerM relative to those of the endomorphism ring Endulcorner(M) having the zero annihilator of each non-zero endomorphism are studied.