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I'm looking for intuition and references for the definition of a hereditary torsion theory and two facts found here. First, the definition and facts:

Definition. A torsion theory $(\mathcal T,\mathcal F$) is hereditary if $\mathcal T$ is closed under subobjects (aswell as quotient objects).

Theorem: In an abelian category $\mathsf A$ satisfying property sup:

  • The assignment $A\mapsto t(A)$ which takes an object into its largest torsion subobject can, using the axiom of choice, be extended to a functor $t:\mathsf A \rightarrow \mathcal T$
  • A torsion theory is hereditary iff $t$ is left exact
  • First of all I have no intuition at all for the notion of hereditary.
  • Is the first part of the theorem mostly technical? What's a reference?
  • What's the intuitive meaning of the left exactness of $t$? When does it have a right adjoint and what would it be? What about an analogous "torsion-free part" functor?

For future reference - does a question like this belong on MSE?

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  • $\begingroup$ The inclusion $i \colon \mathcal T \hookrightarrow \mathsf A$ is a left adjoint of $t$. Did you mean to ask about a right adjoint? $\endgroup$ Feb 1, 2015 at 20:40
  • $\begingroup$ @DagOskarMadsen yes, sorry. Corrected. $\endgroup$
    – Exterior
    Feb 1, 2015 at 22:29

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