I have to prove the following: if $R$ is a unital ring ( non necessarily commutative), that is connected (the only idempotents are 0 and 1) and artinian, then $\frac{R}{J(R)}$ is a domain.
My attempt: Since $R$ is artinian, it is semiperfect. Then $\frac{R}{J(R)}$ is semi-simple and idempotents of $\frac{R}{J(R)}$ lift modulo $J(R)$. Since $R$ is connected, it follows that also $\frac{R}{J(R)}$ is connected. Then I'm stuck. Any hint would be appreciated. Thank you !
Ok, then: after the Artin-Wedderburn theorem, $\frac{R}{J(R)}$ is a product of $n$ matrix ring over division rings. Since the product of at least 2 rings contains non trivial idempotents, (i.e. $(1,0)$), and since $\frac{R}{J(R)}$ is connected, it must be $n=1$. Now, for $N\geq2$ (where $N$ is the size of the matrix), every matrix ring over a division ring contains the non trivial idempotent $(0,0),(0,1)$. Again, by connectivity of $\frac{R}{J(R)}$, we have that $N=1$, so $\frac{R}{J(R)}$ is division ring, hence a domain.