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

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一般类别中的预扭曲理论

我们为研究一般类别的扭转理论提供了一个环境。这个想法是将类别 $\mathcal C$ 中的任何一对 ($\mathcal T$, $\mathcal F$) 完整的完整子类别关联起来，对应的完整子类别 $\mathcal Z = \mathcal T \cap \ $\mathcal C$ 中 \emph {trivial objects} 的 mathcal F$。通过 $\mathcal Z$ 分解的态射称为 $\mathcal Z$-trivial，它们形成了态射的理想，关于它可以定义 $\mathcal Z$-prekernels, $\mathcal Z$-precokernels , 和短的 $\mathcal Z$-preexact 序列。这自然会引出预扭理论的概念，这是本文研究的对象，它包含了阿贝尔语境中的经典理论，即 $\mathcal Z$ 被简化为 $\mathcal C$ 的 $0$ 对象。我们研究预扭曲理论的基本性质，