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A module $M_{R}$ is called semi-artinian if every nonzero image of $M$ contains a simple submodule. Given $m\in M$ and $a_1,a_2,...$ in J(R). Why $ma_1a_2...a_{n-1}a_n=0$ for some $n\geq 1$.


(Edit by KennyTM: The above is OP's original question. The latest, completely changed question follows:)

if $R$ is regular what is the relation between:

1) J(R)

2) a left $R$-module has a projective cover?

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    @Arash: Please visit the meta thread quoted above and explain why you made the radical edits to the questions. Current consensus is to roll-back your edits if you log in and don't address the problem, or don't log in by the end of today.2010-12-15

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This is proved in Askar A. Tuganbaev' s Semidistributive modules and rings, page 153, which you can read on googlebooks.

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    @Matt: I don't seem to be able to get rid of that induction. The idea of the socle filtration is very natural...2010-12-14