Let $R$ be a ring and $M$ be a $R$-module. Let $rad(M)$ be the radical of $M$, that is, the intersection of all maximal submodules of $M$. Moreover, let $soc(M)$ be the socle of $M$, that is, the sum of all simple submodules of $M$. 


We know that both rad and soc define covariant subfunctors of $Id:Mod\_R\rightarrow Mod\_R$. Do radical and socle functors admit left or right adjoints? Thanks in advance for answers.



Adjoints for radical and socle functors
While it does not work for general rings, for Artin algebras one has that the left adjoint of the socle functor is the functor $M \rightarrow M/rad(M)$. I would think that for general rings that is the only choice in case a left adjoint exists.


