ABSTRACT
From this proposition, [2, Theorem 2.2] is clear. Proposition 2.2. If R has only finitely many left ideals, then R/J(K) is the direct sum of finitely many division rings and a finite ring. Proof. By hypothesis, R/J(R) is a semisimple artinian ring, and hence R/J(R) is the direct sum of finitely many simple artinian rings. Let S be one of those simple artinian rings. By Wedderburn structure theorem, S is isomorphic to the full matrix ring Mn(D) with some division ring D and some natural number n. It is easy to see that if D has an infinite number of elements and if n > 1, then Mn(D) has infinitely many left ideals. Therefore, in case n > 1, D must be a finite field. This proves that R/J(R) is a finite direct sum of division rings and finite simple rings. n
Proposition 2.3. If R has only finitely many left ideals, then pR has no subfactor which is isomorphic to T © T, where T is a simple module with infinitely many elements.
