Let $R$ be a finite dimensional algebra and $V$ a R-module. Denote the radical of $V$ by $Rad(V)$ which is defined to be the intersection of all maximal submodules of $V$. The socle of $V$ is the sum of all the irreducible submodules of $V$, and is written $Soc(V)$. The smallest integer $n$ such that $Rad^{n}(V)=0$ is called the Loewy length of the module $V$. Now, suppose that $V$ is indecomposable of Loewy length $2$, that is $Rad^{2}(V)=0$ . I am trying to chow that $Rad(V)=Soc(V)$. Since $Rad^{2}(V)=0$ it follows that $Rad(V)$ is semisimple and then $Rad(V)$ is a submodule of $Soc(V)$. However, the converse is not clear for me.
I would appreciate any hints and comments. Thank you in advance!