Abstract
If R R is right and left noetherian, primitive factor rings are artinian, and R R is right fully bounded, then a simple proof is given to show that finitely generated essential extensions of right artinian modules are artinian. An immediate corollary is that ∩ n = 1 ∞ J n = 0 \cap _{n = 1}^\infty {J^n} = 0 for such a ring.
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