By a noncommutative ring I mean that it has no unit.
I know that if some ring (say, $R$) is simple, then:
$R^2 \neq (0)$
It only possesses $2$ two-sided ideals, namely $(0)$, and itself.
And that, the Jacobson ideal of $R$ is a two-sided ideal of it. So it can either be $0$, or $R$. Now that I want to prove $R$ is semiprimitive, i.e., I have to prove $J(R) = 0$, i.e. to prove that $J(R) \neq R$ (Herstein's book defines $J(R)$ to be $R$ if there's no maximal regular right ideal of $R$), or, in other words, what I need to prove is there exists some maximal regular right ideal of $R$. How can I use the fact that $R$ is Artinian to do so?
Thank you guys very much,
Have a good day, :*
Very early on page $8$ in Noncommutative Rings (also by Herstein), the author notes that irreducible (which I'm reading as 'simple') right $R$ modules all have the form $R/K$ where $K$ is a maximal and regular right ideal. I imagine he includes something similar in the text you are reading.
The Artinian condition, along with $R^2\neq \{0\}$, gives us the existence of simple right $R$ modules (in the form of minimal right ideals).
Can you take it from here?