Sub-exact sequence of quotient modules

Given right module M over ring R and S is a set of the right denominator of M, then Ms is a right quotient module of M over S. Next, given a sequence A →B→C, then we say that the sequence is X-sub-exact if A →X→C exact where X sub-modules of B. If KS-1 is a submodule of MS-1, then we can construct a sub-exact sequence using K.S-1, M.S-1 and M.S-1K.S-1. We can construct a sub-exact sequence for a particular case when R is a division ring. The aim of this paper is to define a sub-exact sequence specifically for quotient modules based on the definition of the sub-exact sequence, define σ (K, L, M) when K, L, M are quotient module, and define maximal element when K, L, M are quotient module. We also construct several examples for sub-exact sequence, σ (K, L, M) and maximal element when the module is replaced by quotient module.