The notion of X-sub-exact sequence of modules is a generalization of exact sequences. Let K, L, M be R-modules and X a submodule of L. The triple (K, L, M) is said to be X-sub-exact at L if K → X → M is exact at X. The exact sequence is a special case of X-sub-exact by taking X = L. We introduce an X-sub-linearly independent module which is a generalization of linearly independent relative to an R-module M by using the concept of X-sub-exact sequence.
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