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Keywords:
$n$-submodule; secondary module; functor; category
$n$-submodule; secondary module; functor; category
Summary:
We explore the existence of $ n $-submodules in the context of module theory. Then we generalize our results by considering an additive, left-exact functor $F$ defined on the category of modules, which is either covariant or contravariant, and preserves multiplications. Within this broader framework, we identify and characterize an $n$-submodule of $F(M)$, derived from the structure of $M$ and the action of the functor $F$.
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