Extending a theorem of Herstein

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).