Much is known about simple rings and these results are often extended to direct sums of simple rings. An alternative approach to extending these results is to loosen the condition simplicity puts on ideals of a ring. Just infinite dimensional algebras are those which, in some sense, contain only few ideals. My talk will begin with examples and a more precise definition of just infinite, after which I will prove the principal results of my dissertation. The first is a theorem that all just infinite algebras are prime; the second result is an example of broadening theorems about simple rings to theorems about just infinite rings.