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?