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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] http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1102911603

[4] http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=3085176&fulltextType=RA&fileId=S0013091500022999

(all of them seem to be available free of charge).

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Just a quick question (or request): could you put the concept of "right semi-artinian" into context, i.e. typical applications of this condition, and what is known about it ? Kind regards - Stephan F. Kroneck. –  bonnbaki Sep 8 '11 at 15:23
    
@bonnbaki: It is hard to give a brief answer. Some usual techniques and main results can be found in the following articles: [1] numdam.org/numdam-bin/fitem?id=BSMF_1968__96__357_0 (in French) [2] ams.org/journals/proc/1971-028-02/S0002-9939-1971-0276259-2/… (all of them seem to be available free of charge). –  Egor Oct 3 '11 at 4:33

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