A "quite superfluous" characterization of the Jacobson radical of an associative ring

It is well-known that the Jacobson radical of a unital ring R is the largest superfluous right ideal of R. While a simple example shows this to be false for non-unital rings, it is shown that the result remains valid in the absence of a unit provided the notion of "superfluous" is taken relative to all regular right ideals. The purpose of this note is to provide a proof for this little-known fact.