In any ring $R$ define the socle as the sum of all minimal right ideals of $R$.
Say we have two minimal ideals $A,B$. If $a\in A,b\in B$, then $a+b$ is in the socle. If $x\in R$, then $(a+b)x=ax+bx$. As $ax\in A$ and $bx\in B$, this element is in $A+B$ and hence in the socle. So the socle is a right ideal.
Is it also a left ideal, and if so, how can I prove it?
If $r\in R$ and $A$ is a minimal right ideal, then $rA$ is either $0$ or a minimal right ideal, because the mapping $R\to R$ defined by $x\mapsto rx$ is a homomorphism of right modules. In any case $rA$ is contained in the sum of the minimal right ideals.