# A left Artinian ring that is also a right Noetherian ring [closed]

I am having trouble showing that a ring which is left Artinian and right Noetherian is right Artinian.

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## closed as off topic by Mark Sapir, Will Jagy, Martin Brandenburg, Bruce Westbury, Tom GoodwillieMar 16 '11 at 2:40

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Elohemahab: are you sure this stuff holds for noncommutative rings? –  darij grinberg Mar 15 '11 at 21:32
For the title: Since [DCC] is stronger than ACC, a left (right) Artinian ring is also a left (right) Noetherian ring. Hence, you can take; for instance, rings for which left and right coincides such as finite rings and commutative rings. –  Unknown Mar 15 '11 at 21:34
That is another homework problem. Voted to close. –  Mark Sapir Mar 15 '11 at 21:45
Hint: Consider the $J$-adic filtration on the ring $A$, where $J$ is the Jacobson radical, and apply Artin-Wedderburn to $A/J$. –  Konstantin Ardakov Mar 15 '11 at 22:33