Given a category $C$ with an orthogonal factorization system $(E,M)$ and a monad $(T,\eta,\mu)$, do the objects $X$ of $C$ for which $\eta_{X} \in M$ form a reflective subcategory?
One obvious idea for constructing the reflection arrow $X \to T'X$ is to consider the $E$-part of the $(E,M)$-factorization $X \to T'X \to TX$ of $\eta_X$. But one also has to show that $\eta_{T'X} \in M$, which I could not do. Please help me!
Alternatively, the above could be false in general, and could then be proven to be true only under additional hypotheses (e.g. $E$ consists of epimorphisms, $M$ consists of monomorphisms, $T$ preserves $E$ (i.e. $e \in E$ implies $Te \in E$), or $T$ preserves $M$ (i.e. $m \in M$ implies $Tm \in M$)).
For a concrete example, let $C$ be the category of abelian groups with its usual (surjection, injection) factorization system and the monad $\mathbb{Q} \otimes_{\mathbb{Z}} -$. Then, the abelian groups $A$ for which $\eta_{A}:A \to \mathbb{Q} \otimes_{\mathbb{Z}} A$ is injective are exactly the torsion-free ones, which are known to form a reflective subcategory. Of course, the image of $\eta_{A}$ is isomorphic to the quotient of $A$ by its torsion subgroup, and in fact, naturally so.
More generally, the above holds for modules over any integral domain $R$ with $\mathbb{Z}$ and $\mathbb{Q}$ replaced by $R$ and its field of fractions respectively. Note that the field of fractions of $R$ is a flat $R$-module (but not faithfully flat unless $R$ was already a field).