  Just infinite algebras have been considered from various perspectives; a
common thread in these treatments is that the notion of just infinite is an
extension of the notion of simple. We reinforce this generalization by
considering some well-known results of Herstein regarding simple rings and
their Lie and Jordan structures and extend these results to their just infinite
analogues. In particular, we prove that if A is a just infinite associative
algebra, of characteristic not 2,3, or 5, then the Lie algebra
$[A,A]/(Z\cap[A,A])$ is also just infinite (where Z denotes the center of A).
