A torsion theory in an abelian category $A$ is a couple $(T,F)$ of additive subcategories called the torsion class $T$ and the torsion free class $F$ such that the following conditions hold:
(in other words, $A(X,Y) = 0$ if $X \in Ob T$ and $Y\in Ob F$).
$Hom(T,Y) = 0 \Rightarrow Y\in Ob F$
$Hom(X,F) = 0 \Rightarrow X\in Ob T$
for all $X\in Ob A$, there exists $Y\subset X$, $Y\in Ob T$ and $X/Y\in Ob F$
Equivalently, a torsion theory in $A$ is a pair $(T,F)$ of strictly full subcategories of $A$ such that the first and last conditions in the above list hold. Alternatively, we can require the last condition and the following 3: $T\cap F=\emptyset$, $T$ is closed under quotients and $F$ under subobjects. It follows also that $T$ and $F$ are stable under extensions.
If the abelian category satisfies the Gabriel’s property (sup) then for every object $X$ there exist the largest subobject $t(X)\subset X$ which is in $T$ and it is called the torsion part of $X$ (sometimes written as $X_T$). Under the axiom of choice, $t: X\to t(X)$ can be extended to a functor.
A torsion theory is hereditary if $T$ is closed under subobjects, or equivalently, $t$ is left exact functor. For some authors (e.g. Golan) torsion theory is assumed to be hereditary.
If $(T,F)$ is a torsion class then $T$ and $F$ both contain the zero object and are closed under biproducts (Borceux II 1.12.3). Presentation of an object $X$ in $Ob A$ as an extension $0\to Y\to X\to X/Y\to 0$, $Y$ in $Ob T$ by $X/Y$ in $Ob F$ is unique up to an isomorphism of short exact sequences (Borceux II 1.12.4).
Given an abelian category $A$ there is a bijection between universal closure operations on $A$, hereditary torsion theories in $A$ (Borceux II 1.12.8) and, if $A$ us locally finitely presentable also with left exact localizations of $A$ admiting a right adjoint and with localizing subcategories of $A$ (Borceux II 1.13.15).
The basic example of a torsion class is the class of torsion abelian groups within the category of all abelian groups. The torsion theories are often used as a means to formulate localization theory in abelian categories.
Comprehensive accounts are in
Historically the notion is introduced in
For a unified treatment in Abelian and triangulated categories see
As explained there, in triangulated context, torsion pairs are in 1-1 correspondence with t-structures. One could also study a relation between torsion theories on an abelian category with tilting theory and $t$-structures on the derived category:
Other references in abelian context include
For analogues in nonadditive contexts see