Some characterizations of hereditarily artinian rings
Published online by Cambridge University Press: 18 May 2009
Throughout this note, rings will mean associative rings with identity and all modules are unital. A ring R is called right artinian if R satisfies the descending chain condition for right ideals. It is known that not every ideal of a right artinian ring is right artinian as a ring, in general. However, if every ideal of a right artinian ring R is right artinian then R is called hereditarily artinian. The structure of hereditarily artinian rings was described completely by Kertész and Widiger  from which, in the case of rings with identity, we get:
A ring R is hereditarily artinian if and only if R is a direct sum S ⊕ F of a semiprime right artinian ring S and a finite ring F.
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