Pretorsion theories in general categories

We present a setting for the study of torsion theories in general categories. The idea is to associate, with any pair ($\mathcal T$, $\mathcal F$) full replete subcategories category C$, corresponding subcategory Z = \mathcal T \cap F$ \emph{trivial objects} C$. morphisms which factor through Z$ are called Z$-trivial, and these form an ideal morphisms, respect one can define Z$-prekernels, Z$-precokernels, short Z$-preexact sequences. This naturally leads notion pretorsion theory, object this article, includes classical abelian context when reduced $0$-object basic properties theories, examine some new examples all endomappings finite sets preordered sets.