A ring $R$ [associative, with 1, not necessary commutative] is said to be right semi-artinian if every non-zero module over $R$ has a simple submodule.
A ring $R$ is said to be strongly $\pi$-regular ($\pi$-regular, right weakly $\pi$-regular) if for every element $x \in R$ there exists an integer $n>0$ such that $x^n \in x^{n+1}R$ (respectively $x^n \in x^n R x^n$, $x^n \in x^n R x^n R$).
Is it true that every right semi-artinian ring must be: 1) strongly $\pi$-regular? 2) $\pi$-regular? 3) right weakly $\pi$-regular? 4) left weakly $\pi$-regular? Or there are some counterexamples?
P.S. Main results on semi-artinian rings can be found in the following articles:
[1] http://www.numdam.org/numdam-bin/fitem?id=BSMF_1968_96_357_0 (in French)
[2] http://www.ams.org/journals/proc/1971-028-02/S0002-9939-1971-0276259-2/home.html
[3] Link
(all of them seem to be available free of charge).