The aim of this paper is to study the concept of intuitionistic L-fuzzy submodules of an R-module M. In this direction, definitions and properties like intersection, sum, product and Cartesian product of two intuitionistic L-fuzzy submodules are defined and discussed. We also introduce the notion of intuitionistic L-fuzzy quotient R-module. The effect of lattice homomorphisms on the image of an intuitionistic L₁-fuzzy submodule A of an R-module M₁ and pre-image of intuitionistic L₂-fuzzy submodule B of an R-module M₂ under module homomorphism are studied. An isomorphism theorem for intuitionistic L-fuzzy modules is also established.

L-fuzzy set, intuitionistic L-fuzzy set, intuitionistic L-fuzzy submodules, intuitionistic L-fuzzy quotient module, lattice isomorphism.