| Summary: | We study the relationship between the category of <i>R</i>-modules (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">C</mi><mrow><mi mathvariant="bold">R</mi><mo>-</mo><mi mathvariant="bold">M</mi></mrow></msub></semantics></math></inline-formula>) and the category of intuitionistic fuzzy modules (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">C</mi><mrow><mi mathvariant="bold">R</mi><mo>−</mo><mi mathvariant="bold">IFM</mi></mrow></msub></semantics></math></inline-formula>). We construct a category <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">C</mi><mrow><mi mathvariant="bold">Lat</mi><mo>(</mo><mi mathvariant="bold">R</mi><mo>−</mo><mi mathvariant="bold">IFM</mi><mo>)</mo></mrow></msub></semantics></math></inline-formula> of complete lattices corresponding to every object in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">C</mi><mrow><mi mathvariant="bold">R</mi><mo>−</mo><mi mathvariant="bold">M</mi></mrow></msub></semantics></math></inline-formula> and then show that, corresponding to each morphism in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">C</mi><mrow><mi mathvariant="bold">R</mi><mo>−</mo><mi mathvariant="bold">M</mi></mrow></msub></semantics></math></inline-formula>, there exists a contravariant functor from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">C</mi><mrow><mi mathvariant="bold">R</mi><mo>−</mo><mi mathvariant="bold">IFM</mi></mrow></msub></semantics></math></inline-formula> to the category <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">C</mi><mi mathvariant="bold">Lat</mi></msub></semantics></math></inline-formula> (=union of all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">C</mi><mrow><mi mathvariant="bold">Lat</mi><mo>(</mo><mi mathvariant="bold">R</mi><mo>−</mo><mi mathvariant="bold">IFM</mi><mo>)</mo></mrow></msub></semantics></math></inline-formula>, corresponding to each object in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">C</mi><mrow><mi mathvariant="bold">R</mi><mo>−</mo><mi mathvariant="bold">M</mi></mrow></msub></semantics></math></inline-formula>) that preserve infima. Then, we show that the category <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">C</mi><mrow><mi mathvariant="bold">R</mi><mo>−</mo><mi mathvariant="bold">IFM</mi></mrow></msub></semantics></math></inline-formula> forms a top category over the category <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">C</mi><mrow><mi mathvariant="bold">R</mi><mo>−</mo><mi mathvariant="bold">M</mi></mrow></msub></semantics></math></inline-formula>. Finally, we define and discuss the concept of kernel and cokernel in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">C</mi><mrow><mi mathvariant="bold">R</mi><mo>−</mo><mi mathvariant="bold">IFM</mi></mrow></msub></semantics></math></inline-formula> and show that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">C</mi><mrow><mi mathvariant="bold">R</mi><mo>−</mo><mi mathvariant="bold">IFM</mi></mrow></msub></semantics></math></inline-formula> is not an Abelian Category.
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