On generalization of exact sequences in modules

The notion of U-exact sequence in modules over rings was introduced in [20] as a generalization of {0}-exact sequence. In this paper, we prove further results on U-exact and V-coexact sequences where V is induced by U. As shown in the commutative diagram fig. 1, wherein if row-1 is U-exact and row-2 is U^′-exact, then we prove that the sequence (0) → ker f^′ → ker f → ker f^′′ is (ker f^′′ ∩U)-exact, and the sequence (0) → coker f^′ → coker f → coker f^′′ is⁽ U^′ +Im f^′′)-exact. We provide explicit examples of the existence of U-exact and U^′-coexactness. Im f^′′.