I'd like to prove the following statement (Exercise 2(i) from Stenstrom, Rings of Quotients, page 157):
Let $\cal C$ be a small abelian category. Then any preradical $r\colon {\cal C}\to{\cal C}$ preserves coproducts.
One has to show that given a fixed object $A\in{\cal C}$, for any family of subobjects $(A_i)_{i\in I}$ of $A$ with embeddings $\varepsilon_i\colon A_i\to \bigoplus_{i\in I}A_i$, then the natural morphism $$ \varrho\colon \bigoplus_{i\in I}r(A_i)\longrightarrow r(\bigoplus_{i\in I}A_i) $$ induced by the family $(r(\varepsilon_i))_{i\in I}$, is an isomorphism.
Now, my idea is to exploit the facts that the preradical $r$ preserves monomorphisms and that, by definition of union of a family of subobjects, namely $\mathop{\rm Im}\varrho=\sum_{i\in I}r(A_i)$, we would have that $$ \bigoplus_{i\in I}r(A_i)=\sum_{i\in I}r(A_i), $$ which would imply that $\varrho$ is a monomorphism, and then that $$ r(\bigoplus_{i\in I}A_i)=\sum_{i\in I}r(A_i) $$ as well. However, I am not able to prove my claims. Can someone help me?
This is not true. For instance, let $\mathcal{C}$ be the opposite category of abelian groups (say, of cardinality at most $2^{\aleph_0}$ to make it essentially small). There is a preradical $r$ on $\mathcal{C}$ which sends each abelian group $A$ to the quotient $A/T(A)$ by the torsion subgroup of $A$. This $r$ does not preserve coproducts (i.e., products of abelian groups); for instance, $r(\mathbb{Z}/(2^n))=0$ for each $n$ but $r(\prod_n\mathbb{Z}/(2^n))$ is nontrivial (for instance, the element of $\prod_n\mathbb{Z}/(2^n)$ which is $1$ on every coordinate is not torsion).