Noetherian semiprimary rings

Question

Is any Noetherian semiprimary ring $R$ Artinian?

By semiprimary I mean $R/J(R)$ semilocal and $J(R)$ nilpotent, where $J(R)$ is the Jacobson radical of $R$.

I know that if $R$ is Artinian then $J(R)$ equals the set $N(R)$ of nilpotent elements of $R$, which is , in the commutative case, the prime radical of $R$. Now, if $P$ is a prime ideal of $R$ which is not a maximal ideal, it falls strictly into a maximal ideal $M$. There exists $m\in M-P$ which is not a nilpotent element, so it does not belong to $J(R)$, so it doesn't to a maximal ideal $K$, and ... .

Answer 1

Even for noncommutative rings,

A semiprimary right Noetherian ring is right Artinian.

By Levitzky's theorem $J(R)$ is a nilpotent ideal, and then by the Hopkins-Levitzki theorem the ring is right Artinian.

Answer 2

If $R/J(R)$ is semilocal, then $R$ is also semilocal. Since $J(R)$ is nilpotent, you are done. (See Atiyah and Macdonald, Corollary 6.11.)