A question about Semisimple ring and their Jacobson radical

I'm reading "Noncommutative Rings" by Herstein, and I got stuck in theorem 1.2.5/page 16. It says: "if $A$ is an two -sided ideal of a noncommutive ring $R$ (may be not unity) then $J(A)=A \cap J(R)$."

Jacobson radical of a ring $R$ is the intersection of all maximal regular right ideal of $R$.

Anyone could explain the theorem?

2 Answers

Let $R$ and $S$ be arbitrary rings and $R$ be a subring of $S$. We know that $\mathrm{rad}S = J$. What can we say about $\mathrm{rad}R$? To answer this question in general is very difficult. But in special cases, for example, when $R$ is two-sided ideal, we can easily find $\mathrm{rad}R$. And this theorem gives the answer how we can do it.

There are two rings (possibly without identity) and there is a Jacobson radical for both. This theorem relates the two in a simple way.

I would imagine what you are reading contains a proof, but if not I would think that switching to the definition of the Jacobson radical via elements would probably be easier.