$ a\,\in\,J(R)\, \, if\, \,and\, \,only\, \,if\, \,Ra\, \,is\, \,a\, \,left\, \, quasi\, \,regular\, \,left \, \,ideal\, \,of \, \,R $
Here $J(R)$ is the Jacobson radical of $R$ where $R$ is a non commutative ring without an identity. I need help with both the necessary and sufficient part. I am unable to show that $ra$ for all $ r\in R$ will be left quasi regular whenever $a$ is and that quasi regularity of $a$ makes $Ra$ left quasi regular
Edit :-
My attempt :-
If $Ra$ is a left quasiregular left ideal of $R$ then $Ra\subseteq J(R)$ $\implies$ $Ra\subseteq I$ for all maximal regular right ideals I of R. Now for all such I there some $e\in R$ specific to that I such that $ (a - ea)\in I$. $Ra\subseteq I$ for all such$ I\implies ea\in I\implies (a - ea + ea) = a\in I\implies a\in J(R) $
Conversely, let $a\in J(R)\implies Ra\subseteq J(R)\,\implies Ra $ is left quasiregular.
Thanks to the comments.