# Ring Theory (semi-artinian)

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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Are you assuming R is noetherian? Not every semiartinian ring is T-nilpotent, that is, not every semiartinian ring is perfect. –  Jack Schmidt Dec 14 '10 at 0:39
Dear Arash, Where did this question come from? The argument that Mariano links to below is quite subtle, and so I'm curious at what level this question was assigned as an exercise (if indeed it was so assigned). Regards, –  Matt E Dec 14 '10 at 2:41
This question completely changed with the edit 3 mins ago. All the above comments and the answer below apply to the previous version, which can be read in the edit history. –  Matt E Dec 14 '10 at 20:34
@Arash: Why are you changing your questions in this manner? Something strange is happening to Arash's questions. I've flagged it for moderator attention (this is not the only question to which this has happened, and some of the previous edits were to silly questions) –  Arturo Magidin Dec 14 '10 at 20:35