Is Jacobson Radical is "non-generating set"?

Question

Denote $R$ ring, $S$ subset. If the two-sided ideal generated by $S$ and the Jacobson radical of $R$ (the set of non-generators of $R$) is $R$, is the two-sided ideal generated by $S$ is $R$?

Answer 1

The Jacobson radical is a collection of "nongenerators" in the following sense:

If $T$ is a right ideal of $R$, and $T+J(R)=R$, then $T=R$.

In other words, the Jacobson radical is a superfluous submodule of $R$.

The same can be said for left ideals.

Of course, what is true of one-sided ideals is also true of two-sided ideals.

Since $\langle S\cup J(R)\rangle=\langle S\rangle +J(R)$, what you described is true.