Geodesic
Some characterizations of hereditarily artinian rings
Glasgow mathematical journal, Tome 28 (1986) no. 1, pp. 21-23

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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 [5] 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. 
Huynh, Dinh van. Some characterizations of hereditarily artinian rings. Glasgow mathematical journal, Tome 28 (1986) no. 1, pp. 21-23. doi: 10.1017/S0017089500006285
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