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Let $R$ be an Artinian Ring and suppose there exists $a,b\in R$ s.t. $I=aR=Rb$, then prove $I=bR=Ra$. (You may assume that a right Artinian Ring is Right Noetherian).

I've managed to get $Ra$, $bR$ contained in $I$ without using the fact that it's Artinian. The hint confuses me because I'm not sure how to make a useful ascending chain to use the fact that it's right Noetherian.

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    Does the ring $R$ have unity?2012-04-26
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    yes it does have unity2012-04-26
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    @Arturo Maybe I´m missing something. But how can you conclude $a=bu$? It seems to me that you only have $a=ub$, and you need somehow to prove that $bR$ is a two-sided ideal to get the conclusion. (In specific the Artinianity hypothesis is missing or well hidden...)2012-04-26
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    @Giovanni: I got some wires crossed. Sorry.2012-04-26
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    Before you go any further looking for ascending and decending chains, see if you can't make use of the *maximal condition* and/or *minimal condition* first.2012-04-28
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    I still can't seem to get anything (and also I don't know how to type maths symbols), my exam is in a few days, if anyone knows the answer please tell me2012-04-29

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