or all x and y in R, and either (R, [R, R], R) = 0, or (R, R, [R, R]) = 0. His main result is that prime rings satisfying these identities must be either commutative or associative. Kleinfeld and Kleinfeld [S] replaced [(x, y, x), y] = 0 in Thedyâs hypotheses with (x, y*, x) = y(x, y, x) + (x, y, x) y. In that case the prime rings of characteristic 22, 3 all must have commutators in the center and the simple rings of this variety must be commutative or associative or belong to a third class, which are quadratic and Lie admissible, but not flexible. In the present paper we study rings which satisfy (i) ([R, R], R, R)=O, and (ii) [(a, b, a), R] =O. Thus our (i) is weaker than Thedyâs hypothesis, but our (ii) is somewhat stronger than his [(x, y, x), y] = 0. We prove that prime rings satisfying (i) and (ii) must be either commutative or associative. These papers are in the tradition of Albert [ 11, to search for the most general identities that suffice for a certain result. In these instances we are in the join of commutative and associative rings. Jordan rings are after all commutative, so we could add the Jordan identity (x2, y, x) = 0 at any time if we desire to switch these