A ring $R$ is called semi-primary if $R/J(R)$ is semisimple and $J(R)$ is nilpotent (with $J(R)$ its Jacobson radical).
I've been trying to find some example of non-artinian semi-primary rings but I can't figure it out.
Is there some book where I can find examples or hint to build-up one of this rare rings?
Consider the ring $K[x_1,x_2,x_3,...]/I$, where $K$ is a field and $I$ is the ideal generated by all products $x_ix_j$ where $i$ and $j$ are (possibly equal) positive integers. Then, this ring is a local ring whose unique maximal ideal $\mathbf{m}$ is the ideal $(x_1,x_2,x_3,...)$, which is nilpotent since the product of any two elements of $\mathbf{m}$ is zero. Of course, the unique maximal ideal is also the Jacobson radical, and the quotient ring is isomorphic to $K$, which is a field and hence semisimple.
Since $\mathbf{m}$ is not a finitely generated ideal, the ring $K[x_1,x_2,x_3,...]/I$ is not Noetherian. Hence, by the Hopkins–Levitzki theorem, it cannot be Artinian either. In fact, the same theorem also shows that any example must also be non-Noetherian.