Intuitionistic L-fuzzy essential and closed submodules

Let R be a commutative ring with identity and M be an R-module. An intuitionistic L-fuzzy submodule (ILFSM) C of an intuitionistic L-fuzzy module A of R-module M, is called an intuitionistic L-fuzzy essential submodule in A, if C ∩ B ≠ χ{θ} for any non-trivial ILFSM B of A. In this case we say that A is an essential extension of C. Also, if C has no proper essential extension in A, then C is called an intuitionistic L-fuzzy closed submodule in A. Further, for ILFSMs B, C of A, C is called complement of B in A if C is maximal with the property that B ∩ C = χ{θ}. We study these mentioned notations which are generalization of the notions of essential submodule, closed submodule and complement of a submodule in the intuitionistic L-fuzzy module theory. We prove many basic properties of both these concepts.